text:
\section{Introduction}
To determine the Hausdorff dimensions of sets of numbers in which the distributions of the digits are of specific characters for some representation is a fundamental and important problem in number theory and multifractal analysis. It has a long history and there are a great many classic work on this topic, suh as Besicovitch~\cite{Be}, Eggleston~\cite{E}, Billingsley~\cite{Bi}, Barreira et al.~\cite{BSS} and Fan et al.~\cite{FF,FFW}, in which the sets are associated with frequencies of digits of numbers in different expansions, long-term time averages and ergodic limits, etc.
Actually, the expressions in these sets are of a common feature, i.e., they are usually described by the arithmetic mean. Different from this, in this paper we would like to study the moving digit means of numbers and investigate the corresponding level sets related to it in the unit interval.

The moving digit mean is derived from the concept of tangential dimension at a point for measure studied by Guido and Isola~\cite{GI1,GI2}. To be precise, if the measure is a Bernoulli measure, then the tangential dimension is of a linear relation with the moving digit mean (see~\cite{CWX}). More significantly, when we explore the multifractal behavior at a point of a measure, the tangential dimension is more sensitive than the local dimensions of measures, which enables the tangential dimension to provide more information than the local dimension. Thus, compared with the arithmetic mean, the moving digit mean used in the description of level sets can also provide more information on the distributions of digits of numbers. In the following, we introduce the corresponding concepts and notations.

Let $p\geq 2$ be an integer and $A=\{0,1,\ldots p-1\}$ be the alphabet with $p$ elements. It is known that each number $x\in I= [0,1]$ can be expanded into an infinite non-terminating expression
\[\sum_{n=1}^\infty\frac{x_n}{p^n}=0.x_1x_2x_3\ldots,\quad\text{where}\ x_n\in A,\ n\geq1,\]
which is called the $p$-adic expansion of $x$. Denoted by $S_n(x)=\sum_{i=1}^nx_i, n\geq1$, the $n$-th partial sum of $x$. Let $T\colon I\to I$ be the shift operator defined by
\[Tx=0.x_2x_3x_4\ldots,\quad\text{for any}\ x=0.x_1x_2x_3\ldots\in I.\]
Let $x\in I$, we call
\begin{align}
  \b{\it M}(x)=\lim_{n\to\infty}\varliminf_{m\to\infty}\frac{S_n(T^mx)}{n}\quad\text{and}\quad\bar{M}(x)=\lim_{n\to\infty}\varlimsup_{m\to\infty}\frac{S_n(T^mx)}{n}
\end{align}
\emph{the lower and upper moving digit means} of $x$ respectively.

\begin{rem}
	Let $n\geq1$. Denote by
	\[\b{\it M}_n(x)=\varliminf_{m\to\infty}\frac{S_n(T^mx)}{n}\quad\text{and}\quad\bar{M}_n(x)=\varlimsup_{m\to\infty}\frac{S_n(T^mx)}{n}\]
	the $n$-th lower and upper moving digit means respectively. It is easy to check that, for any $x\in I$, the two sequences $\{-\b{\it M}_n(x)\}_{n\geq1}$ and $\{\bar{M}_n(x)\}_{n\geq1}$ are both subadditive since they satisfy  the inequalities:
	\[-\b{\it M}_{m+n}(x)\leq-\b{\it M}_m(x)-\b{\it M}_n(x)\quad\text{and}\quad\bar{M}_{m+n}(x)\leq\bar{M}_m(x)+\bar{M}_n(x),\quad m,n\geq1.\]
	Thus, the limits of $\b{\it M}_n(x)$ and $\bar{M}_n(x)$ always exist as $n\to\infty$. So, the definitions of $\b{\it M}(x)$ and $\bar{M}(x)$ are both reasonable.
\end{rem}

To investigate the influence of the lower and upper moving digit means on the distribution of digits of numbers, define the level set
\begin{align}
	B(\alpha,\beta)=\{x\in I\colon \b{\it M}(x)=\alpha,\bar{M}(x)=\beta\},\quad\text{where}\ 0\leq\alpha\leq\beta\leq p-1,
\end{align}
which may be called \emph{Banach set} with lower level $\alpha$ and upper level $\beta$. In this paper, we would like to determine the Hausdorff dimension of set $B(\alpha,\beta)$ for any $p\geq2$, which is a non-trivial and meaningful generalization of the work in \cite{CWX}. First, we give the definition of $p$-adic entropy function below.

Let $0\leq\alpha\leq p-1$ and $\delta>0$. Denote
\[H(\alpha,n,\delta)=\left\{x_1x_2\cdots x_n\in A^n\colon n(\alpha-\delta)<\sum_{i=1}^nx_i<n(\alpha+\delta)\right\}\]
and $h(\alpha,n,\delta)=\card H(\alpha,n,\delta)$. Here and in the sequel, the symbol $\card$ is the cardinality of some finite set. Furthermore, define the $p$-adic entropy function as
\begin{align}\label{definiton h alpha}
	h(\alpha)=\lim_{\delta\to0}\varlimsup_{n\to\infty}\frac{\log h(\alpha,n,\delta)}{(\log p)n},\quad 0\leq\alpha\leq p-1.
\end{align}
It is evident that $0\leq h(\alpha)\leq1$ for any $0\leq\alpha\leq p-1$.

Denote by $\dim_H$ the Hausdorff dimension of some set. Then we have
\begin{thm}\label{theorem main theorem}
	For any $0\leq\alpha\leq\beta\leq p-1$, we have
\begin{align}\label{formula main}
	\dim_HB(\alpha,\beta)=\sup_{\alpha\leq t\leq\beta}h(t).
\end{align}
\end{thm}

In particular, let $0\leq\alpha\leq p-1$ and write
\begin{align}
	B(\alpha):=B(\alpha,\alpha)=\{x\in I\colon \b{\it M}(x)=\bar{M}(x)=\alpha\}.
\end{align}
Then we may obtain immediately that
\begin{cor}\label{corollary main corollary}
	For any $0\leq\alpha\leq p-1$, we have $\dim_HB(\alpha)=h(\alpha)$.
\end{cor}

Note that we will discuss another set $B^\ast(\alpha)$ which is related to the moving digit mean $M(x)$ in the end of this paper. As a result, $B(\alpha)$ is   different from and large than $B^\ast(\alpha)$ in the sense of Hausdorff dimension.

Moreover, in the case $p=2$, we can give an explicit expression to the binary entropy function $h_2$ by the following calculation:
\begin{align}
	  h_2(\alpha)=\lim_{\delta\to0}\varlimsup_{n\to\infty}\frac{\log \sum_{i=[n(\alpha-\delta)]+1}^{[n(\alpha+\delta)]}\binom{n}{i}}{(\log 2)n}=\frac{-\alpha\log\alpha-(1-\alpha)\log(1-\alpha)}{\log2},
\end{align}
where $\binom ni$ is the binomial coefficient. Thus, the following conclusion can be easily obtained by Theorem \ref{theorem main theorem} and Corollary \ref{corollary main corollary}.

\begin{cor}
	Let $p=2$. If $0\leq\alpha\leq\beta\leq 1$, then $\dim_HB(\alpha,\beta)=\sup_{\alpha\leq t\leq\beta}h_2(t)$. If $0\leq\alpha\leq1$, then $\dim_HB(\alpha)=h_2(\alpha)$.
\end{cor}

In the present paper, the readers are assumed to know well the definitions and basic properties of Hausdorff dimension and Hausdorff measure. For these and more related theory, one can refer to Falconer's book~\cite{F97}.

The structure of this paper is as follows. In the next section, the concepts of $p$-adic lower and upper entropy functions are introduced, their relations to the $p$-adic entropy function are also shown. In Section 3, the concepts Besicovitch sets in $p$-adic expansion are introduced. Moreover, their Hausdorff dimensions are also determined, which generalizes an early work of Besicovitch. A In Section 4, we will introduce some special Moran sets and determine their Hausdorff dimension for ready use. The last section is devoted to the proof of Theorem \ref{theorem main theorem}, some further discussion are also presented there.

\section{$p$-adic entropy function}
In this section, we will introduce the definitions of $p$-adic lower entropy function $\b{\it h}(\alpha)$ and $p$-adic upper entropy function $\bar{h}(\alpha)$, and then present some properties about the two functions and the $p$-adic entropy function $h(\alpha)$.

Let $0\leq\alpha\leq p-1$, $n\geq1$ and $\delta>0$. Denote
\[\bar{H}(\alpha,n,\delta)=\left\{x_1x_2\cdots x_n\in A^n\colon\sum_{i=1}^nx_i<n(\alpha+\delta)\right\}\]
and $\bar{h}(\alpha,n,\delta)=\card\bar{H}(\alpha,n,\delta)$. Define the $p$-adic upper entropy function
\begin{align}\label{definition bar h}
\bar{h}(\alpha)=\lim_{\delta\to0}\varlimsup_{n\to\infty}\frac{\log\bar{h}(\alpha,n,\delta)}{(\log p)n}.
\end{align}
Oppositely, denote
\[\b{\it H}(\alpha,n,\delta)=\left\{x_1x_2\cdots x_n\in A^n\colon\sum_{i=1}^nx_i>n(\alpha-\delta)\right\}\]
and $\b{\it h}(\alpha,n,\delta)=\card\b{\it H}(\alpha,n,\delta)$. Define the $p$-adic lower entropy function
\begin{align}\label{definition b h}
  \b{\it h}(\alpha)=\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log\b{\it h}(\alpha,n,\delta)}{(\log p)n}.
\end{align}
Note that we have $0\leq\b{\it h}(\alpha),\bar{h}(\alpha)\leq1$ and the limits in \eqref{definition bar h} and \eqref{definition b h} both exist since $\bar{h}(\alpha,n,\delta)$ and $\b{\it h}(\alpha,n,\delta)$ are increasing for $\delta>0$. Moreover, for the two functions $\b{\it h}(\alpha)$ and $\bar{h}(\alpha)$, we have the following relation between them.
\begin{thm}\label{theorem barh bh}
	$\bar{h}(p-1-\alpha)=\b{\it h}(\alpha)$.
\end{thm}
\begin{proof}
For any $n$-word $x_1\cdots x_n\in A^n$, it is easy to check that the word $(p-1-x_1)\cdots(p-1-x_n)\in\b{\it H}(\alpha)$ if and only if the word $x_1\cdots x_n\in\bar{H}(p-1-\alpha)$. So, there is a one-to-one corresponding between the two sets $\b{\it H}(\alpha)$ and $\bar{H}(p-1-\alpha)$.
\end{proof}

Recall the definition of $p$-adic function $h(\alpha)$ in \eqref{definiton h alpha}. Actually, we may have
\begin{align}\label{definiton h alpha ==}
h(\alpha)=\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log h(\alpha,n,\delta)}{(\log p)n}=\lim_{\delta\to0}\varlimsup_{n\to\infty}\frac{\log h(\alpha,n,\delta)}{(\log p)n}
\end{align}
according to Proposition 4.2 in~\cite{TWWX}. Moreover, we can also discover the following properties of $p$-adic function.

\begin{thm}\label{theorem enumerate 1}
	For the function $h(\alpha)$ defined on $[0,p-1]$, we have
\noindent
    \begin{enumerate}
    	\item\label{halpha 1} $h(0)=h(p-1)=0$;
		\item\label{halpha 2} $h(\alpha)$ is concave and continuous on $[0,p-1]$;
		\item\label{halpha 3} $h(\alpha)$ is symmetric with respect to the line $\alpha=(p-1)/2$. That is, we have $h(\alpha)=h(p-1-\alpha)$ for any $0\leq\alpha\leq p-1$. It follows that $h(\alpha)$ is increasing on $[0,(p-1)/2]$ and decreasing on $[(p-1)/2,p-1]$;
		\item\label{halpha 4} If $0\leq\alpha\leq(p-1)/2$, then $\bar{h}(\alpha)=h(\alpha)$; if $(p-1)/2\leq\alpha\leq p-1$, then $\b{\it h}(\alpha)=h(\alpha)$.
	\end{enumerate}
\end{thm}
\begin{proof}
(1) The conclusion $h(0)=0$	is followed by the the estimation
\begin{align}\label{inequality h zero}
   h(0)\leq\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log\frac{n^{[n\delta]}}{[n\delta]!}}{(\log p)n}\leq\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log\frac{n^{n\delta}}{(n\delta-1)!}}{(\log p)n}		=\lim_{\delta\to0}\frac{\delta\log\frac{e}{\delta}}{\log p}=0.
\end{align}
Here, the third equality is followed by the well-known Stirling's approximation and $y!=y(y-1)\cdots(y-[y])$ if $y>0$. The other conclusion $h(p-1)=0$ can be deduced similarly.

(2) Let $m\geq1$ and take $m$ $n$-words $X_1,X_2,\ldots,X_m\subset H(\alpha,n,\delta)$. It is obvious that the concatenation of these words satisfies $X_1X_2\cdots X_m\subset H(\alpha,nm,\delta)$. Thus, \[\big(h(\alpha,n,\delta)\big)^m\leq h(\alpha,nm,\delta).\]
Let $\alpha,\beta\in (0,p-1)$ and $s$, $t$ be two positive integers. Then
\[\big(h(\alpha,n,\delta)\big)^s\big(h(\beta,n,\delta)\big)^t\leq h(\alpha,ns,\delta)h(\beta,nt,\delta)\leq h\left(\frac{s\alpha+t\beta}{p+q},n(s+t),\delta\right).\]
Hence,
\[\frac{s}{s+t}h(\alpha)+\frac{t}{s+t}h(\beta)\leq h\left(\frac{s}{s+t}\alpha+\frac{t}{s+t}\beta\right).\]
Since $s$ and $t$ are two arbitrary positive integers, we have
\[\lambda h(\alpha)+(1-\lambda)h(\beta)\leq h(\lambda\alpha+(1-\lambda)\beta)\]
for any $0<\lambda<1$. It means that the function $h(\alpha)$ is rational concavity.

By the definition of $h(\alpha)$, for any $\eta>0$, there exists $\delta_0>0$ such that
\begin{align}\label{inequality h alpha}
\varlimsup_{n\to\infty}\frac{\log h(\alpha,n,\delta)}{(\log p)n}<h(\alpha)+\frac{\eta}{2}
\end{align}
for any $0<\delta<\delta_0$.
Take a number $\gamma$ which satisfies $|\alpha-\gamma|<\delta/2$. Then, we have $H(\gamma,n,\delta/2)\subset H(\alpha,n,\delta)$. It yields that $h(\gamma,n,\delta/2)\leq h(\alpha,n,\delta)$. Moreover, by the definition of $h(\gamma)$, there exists some $\delta_1$ satisfying $0<\delta_1<\delta_0$ such that
\[h(\gamma)\leq \varlimsup_{n\to\infty}\frac{\log h(\gamma,n,\delta/2)}{(\log p)n}+\frac{\eta}{2}\leq\varlimsup_{n\to\infty}\frac{\log h(\alpha,n,\delta)}{(\log p)n}+\frac{\eta}{2}.\]
for any $0<\delta<\delta_1$. This, together with \eqref{inequality h alpha}, yields that
$h(\gamma)<h(\alpha)+\eta$ if $|\alpha-\gamma|<\delta/2$, where  $0<\delta<\delta_1$.
It implies that the function $h(\alpha)$ is upper semi-continuous. So, $h(\alpha)$ is concave and then continuous on $(0,p-1)$.

Next, we show that $h(\alpha)$ is continuous at $\alpha=0$ and $\alpha=p-1$. Since
$h(\alpha,n,\delta)\leq\bar{h}(\alpha,n,\delta)$, similar to the estimation \eqref{inequality h zero} we have
\begin{align*}
  \lim_{\alpha\to0^+}h(\alpha)&\leq\lim_{\alpha\to0^+}\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log\big(n^{[n(\alpha+\delta)]}/[n(\alpha+\delta)]!\big)}{(\log p)n}\\
  &\leq\lim_{\alpha\to0^+}\lim_{\delta\to0}\frac{(\alpha+\delta)\log\frac{e}{(\alpha+\delta)}}{\log p}=\lim_{\alpha\to0^+}\frac{\alpha\log\frac{e}{\alpha}}{\log p}=0.
\end{align*}
It follows that $\lim_{\alpha\to0^+}h(\alpha)=0=h(0)$. Thus, $h(\alpha)$ is continuous at $\alpha=0$. Similarly, the continuity of $h(\alpha)$ at $\alpha=p-1$ holds as well. Thus, $h(\alpha)$ is concave and continuous on $[0,p-1]$.

(3) The proof of the property $h(\alpha)=h(p-1-\alpha)$ is similar to the discussion for Theorem~\ref{theorem barh bh}. The monotonicity of $h(\alpha)$ is followed by the concavity of $h(\alpha)$ in property \eqref{halpha 2}.

(4) The former part is followed by the increasing property of $h(\alpha)$ in \eqref{halpha 3} and the inequality \[h(\alpha,n,\delta)\leq\bar{h}(\alpha,n,\delta)\leq2\left(\left[\frac{\alpha+\delta}{2\delta}\right]+1\right)h(\alpha,n,\delta),\]
where $\alpha\leq(p-1)/2$. The second part can be dealt with in a similar way.
\end{proof}

\section{Besicovitch sets}
In this section, we will determine the Hausdorff dimensions of Bsicovitch sets $\b{\it E}(\alpha)$ and $\bar{E}(\alpha)$ and present at last a further property of the $p$-adic entropy function for ready use.

Let $x\in I$. Denote by
\[\b{\it A}(x)=\varliminf_{n\to\infty}\frac{S_n(x)}{n}\quad\text{and}\quad \bar{A}(x)=\varlimsup_{n\to\infty}\frac{S_n(x)}{n}\]
the lower and upper digit means of $x$, respectively. Let $0\leq\alpha\leq p-1$. Define the level sets
\begin{align}\label{E alpha}
   \b{\it E}(\alpha)=\left\{x\in I\colon \b{\it A}(x)\geq\alpha\right\}\quad\text{and}\quad\bar{E}(\alpha)=\left\{x\in I\colon \bar{A}(x)\leq\alpha\right\},
\end{align}
which are called the \emph{Besicovitch sets} in this paper. Note that the Hausdorff dimensions of $\b{\it E}(\alpha)$ and $\bar{E}(\alpha)$ are determined by Besicovitch~\cite{Be} in the case of binary expansion. For the present general case, we have

\begin{thm}\label{theorem general}
	Let $0\leq\alpha\leq p-1$. Then
	\begin{equation}\label{equation E alpha 1}
	\dim_H\b{\it E}(\alpha)=\begin{cases} 1,\indent &0\leq\alpha<(p-1)/2;\\
	h(\alpha),\indent &(p-1)/2\leq\alpha\leq p-1,\end{cases}
	\end{equation}
	and
	\begin{equation}\label{equation E alpha 2}
    \dim_H\bar{E}(\alpha)=\begin{cases} h(\alpha),\indent &0\leq\alpha<(p-1)/2;\\
    1,\indent &(p-1)/2\leq\alpha\leq p-1.\end{cases}
    \end{equation}
\end{thm}

To prove this theorem, we would like to introduce a lemma about the Hausdorff dimensions of homogeneous Moran sets. Here, It is assumed the readers are familiar with the definition and structure of homogeneous Moran sets, for which one can see \cite{FWW} for more details.

Let $\{N_k\}_{k\geq1}$ be a sequence of integers and $\{c_k\}_{k\geq1}$ be a sequence of positive numbers satisfying $N_k\geq2$, $0<c_k<1$ and $N_kc_k\leq1$. Let $\mathcal{M}=\mathcal{M}\big(I,\{N_k\}_{k\geq1},\{c_k\}_{k\geq1}\big)$ be the homogeneous Moran set determined by the sequences $\{N_k\}_{k\geq1}$ and $\{c_k\}_{k\geq1}$. Denote
\[s=\varliminf_{k\to\infty}\frac{\log(N_1N_2\cdots N_k)}{-\log(c_1c_2\cdots c_{k+1}N_{k+1})}.\]
Then we have
\begin{lem}[See Theorem 2.1 and Corollary 2.1 in \cite{FWW}]\label{lemma FWW}
	Let $\mathcal{M}$ be a homogeneous Moran set, then $\dim_H\mathcal{M}\geq s$. Moreover, if $\inf_{k\geq1}c_k>0$, then $\dim_H\mathcal{M}=s$.
\end{lem}

\begin{proof}[Proof of Theorem \ref{theorem general}]
We will give only the proof of \eqref{equation E alpha 2} since the conclusion \eqref{equation E alpha 1} can be dealt with in a similar way.

For the second part of \eqref{equation E alpha 2}, it is obvious that
$\{x\in I\colon\lim_{n\to\infty}S_n(x)/n=(p-1)/2\}\subset\bar{E}(\alpha)$. Since $\lim_{n\to\infty}S_n(x)/n=(p-1)/2$ for almost all $x\in I$ by the ergodic theorem, we have
\[\dim_H\bar{E}(\alpha)\geq\dim_H\left\{x\in I\colon\lim_{n\to\infty}\frac{S_n(x)}{n}=\frac{p-1}{2}\right\}=1.\]
It follows that $\dim_H\bar{E}(\alpha)=1$ as $(p-1)/2\leq\alpha\leq p-1$.

For the first part of \eqref{equation E alpha 2}, first we will show the upper bound of Hausdorff dimension of $\bar{E}(\alpha)$ is $h(\alpha)$ as $0\leq\alpha<(p-1)/2$. For any $\delta>0$, we have
\[\bar{E}(\alpha)\subset\bigcap_{l=1}^\infty\bigcup_{n=l}^\infty\bigcup_{x_1\cdots x_n\in\bar{H}(\alpha,n,\delta)}I(x_1\cdots x_n),\]
where the cylinder $I(x_1\cdots x_n)=\{y=0.y_1y_2\ldots\in I\colon y_1=x_1,\ldots,y_n=x_n\}$. By the definition of $\bar{h}(\alpha)$, for any $\eta>0$, there exists an integer $N$ such that
\[\bar{h}(\alpha,n,\delta)<p^{n\left(\bar{h}(\alpha)+\frac{\eta}{2}\right)},\quad\forall n>N.\]
Then, for any $l>N$, the $(\bar{h}(\alpha)+\eta)$-Hausdorff measure of $\bar{E}(\alpha)$ satisfies
\[\mathbb{H}_{p^{-l}}^{\bar{h}(\alpha)+\eta}\big(\bar{E}(\alpha)\big)\leq\sum_{n=l}^{\infty}\bar{h}(\alpha,n,\delta)(p^{-n})^s<\sum_{n=l}^{\infty}(p^{-\frac{\eta}{2}})^n<\infty.\]
This implies that $\dim_H\bar{E}(\alpha)\leq\bar{h}(\alpha)+\eta$. Thus, $\dim_H\bar{E}(\alpha)\leq\bar{h}(\alpha)=h(\alpha)$ by the arbitrariness of $\eta$ and \eqref{halpha 4} of Theorem \ref{theorem enumerate 1}.

Next, we turn to show that the lower bound of the Hausdorff dimension of $\bar{E}(\alpha)$ is $h(\alpha)$. For this, we will prove $\dim_H\bar{E}(\alpha)\geq\tau$ for any $0<\tau<h(\alpha)$.

Since $\tau<h(\alpha)$, we can take two sequences, one is an increasing integer sequence $\{n_j\}_{j\geq1}$ and the other is a decreasing positive sequence $\{\delta_j\}_{j\geq1}$ satisfying $\lim_{j\to\infty}\delta_j=0$, such that
\[h(\alpha,n_j,\delta_j)>p^{n_j\tau}.\]
Let $j\geq1$ and write
\[F_j(\alpha)=\left\{x_1x_2\cdots x_{n_j}\in A^{n_j}\colon\left|\frac{\sum_{i=1}^{n_j}x_i}{n_j}-\alpha\right|<\delta_j\right\}.\]
Take a positive integer sequence $\{m_i\}_{i\geq1}$ satisfies
\[\lim_{j\to\infty}\frac{n_{j+1}}{\sum_{i=1}^{j}m_in_i}=0.\] Denote by \[q_j=m_1n_1+m_2n_2+\ldots+m_jn_j,\quad j\geq1.\]
Based on the sequence of sets $\{F_j(\alpha)\}_{j\geq1}$, construct the Moran set
\begin{align*}
	\begin{split}
		\mathcal{F}(\alpha)&=\left\{0.x_1x_2\ldots\in I\colon x_{q_i+1}\cdots x_{q_{i+1}}\in F_i(\alpha)^{m_i},\forall i\geq1\right\}\\
		&=:0.\prod_{i=1}^{\infty}F_i(\alpha)^{m_i}.
	\end{split}
\end{align*}
Here and in the sequel, if $F$ is a set of words with equal length and $m$ is a positive integer, then we use the notation $F^m$ to denote the set in which every word is the concatenations of $m$ words in the set $F$, and $F^\infty$ the set in which every sequence is the concatenations of infinite words in the set $F$. For the sequence of sets of words $\{F_i\}_{i\geq1}$,   $\prod_{i=1}^\infty F_i$ denotes the set in which every sequence is the successive concatenations of words in the set $F_i$ according to the order of natural numbers.

It is easy to see that
\[\frac{m_1n_1(\alpha-\delta_1)+\cdots+m_jn_j(\alpha-\delta_j)}{m_1n_1+\cdots+m_jn_j}\leq\frac{S_{q_j}(x)}{q_j}\leq\frac{m_1n_1(\alpha+\delta_1)+\cdots+m_jn_j(\alpha+\delta_j)}{m_1n_1+\cdots+m_jn_j}.\]
Since $m_jn_j\to\infty$ and $\delta_j\to0$ as $j\to\infty$, we have
\[\lim_{j\to\infty}\frac{S_{q_j}(x)}{q_j}=\alpha\]
by the squeeze theorem. This implies the upper limit of $S_n(x)/n$ is $\alpha$. So, we have $\mathcal{F}(\alpha)\subset\bar{E}(\alpha)$ and then $\dim_H\bar{E}(\alpha)\geq\dim_H\mathcal{F}(\alpha)$.

For any integer $n$ large enough, there exist two integers $j\geq1$ and $b$ such that
\[0\leq b<m_{k+1}\quad\text{and}\quad\sum_{i=1}^jm_in_i+bn_{j+1}\leq n<\sum_{i=1}^jm_in_i+(b+1)n_{j+1}.\]
Then, by the first assertion of Lemma \ref{lemma FWW} we have
\[\dim_H\mathcal{F}(\alpha)\geq\varliminf_{j\to\infty}\frac{\big(\sum_{i=1}^jm_in_i+bn_{j+1}\big)\tau\log p}{\big(\sum_{i=1}^jm_in_i+(b+1)n_{j+1}\big)\log p-n_{j+1}\tau\log p}=\tau.\]
Thus, we obtain that $\dim_H\bar{E}(\alpha)\geq\tau$, which shows the first part of \eqref{equation E alpha 2}.

The proof is completed now.
\end{proof}

Moreover, denote by
\[A(x)=\lim_{n\to\infty}\frac{S_n(x)}{n},\quad x\in I,\]
\emph{the arithmetic digit mean} of $x$ if the limit exists and define the level set related to it as
\begin{align}\label{E = alpha}
	E(\alpha)=\left\{x\in I\colon A(x)=\alpha\right\}.
\end{align}
Then, by the same technique used in the proof of Theorem~\ref{theorem general}, we may get

\begin{thm}\label{theorem = general}
	For any $0\leq\alpha\leq p-1$, we have that $\dim_HE(\alpha)=h(\alpha)$.
\end{thm}

\begin{cor}\label{corollary p-1 2}
  $h\big((p-1)/2\big)=1$.
\end{cor}
\begin{proof}
Since $A(x)=(p-1)/2$ for almost all $x\in I$, by Theorem~\ref{theorem = general} we have
\begin{align*}
	\begin{split}
		1=\dim_H\left\{x\in I\colon A(x)=\frac{p-1}{2}\right\}
		=\dim_HE\left(\frac{p-1}{2}\right)=h\left(\frac{p-1}{2}\right).
	\end{split}
\end{align*}
It ends the proof.
\end{proof}

\begin{rem}\label{remark b alpha beta}
	By Corollary \ref{corollary p-1 2}, Theorem \ref{theorem main theorem} can be restated in details as follows: if $0\leq\alpha\leq\beta<(p-1)/2$, then $\dim_HB(\alpha,\beta)=h(\beta)$; if $0\leq\alpha\leq(p-1)/2\leq\beta\leq1$, then $\dim_HB(\alpha,\beta)=1$; if $(p-1)/2<\alpha\leq\beta\leq p-1$, then $\dim_HB(\alpha,\beta)=h(\alpha)$.
\end{rem}

\section{Some Moran sets}
In this section, we will introduce some Moran sets constructed by sets of words with bounded digit sums and then determine their Hausdorff dimensions. Based on them, we will construct suitable subsets to achieve the lower bound of Hausdorff dimension of $B(\alpha,\beta)$ in the last section.

Let $M\geq1$ be an integer. Take two integers $P$ and $Q$ satisfying $0\leq P\leq Q\leq (p-1)M$. Write
\[W([P,Q],M):=\left\{x_1x_2\cdots x_M\in A^M\colon P\leq\sum_{i=1}^Mx_i\leq Q\right\}.\]
Then define the Moran set
\begin{align*}
    \mathcal {W}([P,Q],M)&:={}0.W([P,Q],M)^\infty.
\end{align*}
For the size of the set $\mathcal {W}([P,Q],M)$, by the second assertion in Lemma \ref{lemma FWW}, we can get immediately that
\begin{lem}\label{lemma pqm}
	Let $0\leq P\leq Q\leq (p-1)M$ and $M\geq1$. Then
	\[\dim_H\mathcal {W}([P,Q],M)=\frac{\log\card W([P,Q],M)}{(\log p)M}.\]
\end{lem}
Here and in the sequel, if $P=Q$, then write $W([P,Q],M)$ as $W(P,M)$ and $\mathcal {W}([P,Q],M)$ as $\mathcal {W}(P,M)$ respectively for simplicity.

Let $\alpha$ be a real number and $0\leq\alpha\leq p-1$. Let $n\geq1$. Define the function
\[\omega(\alpha,n)=\card W([\alpha n],n).\]
Then the corresponding properties in the following lemma is evident.

\begin{lem}\label{theorem enumerate 2}
   Let $0\leq\alpha\leq p-1$ and $n\geq1$. Then
\begin{enumerate}
	\item\label{omega 1} For each $\alpha$, $\omega(\alpha,n)$ is increasing with respect to $n$;
	\item\label{omega 2} For each $n$, $\omega(\alpha,n)$ is constant on      $[(k-1)/n,k/n)$, where $1\leq k\leq(p-1)n$, with respect to $\alpha$;
    \item\label{omega 3} For each $n$, $\omega(\alpha,n)$ is increasing on $[0,(p-1)/2]+1/n)$ and decreasing on $[(p-1)/2+1/n,p-1]$ with respect to $\alpha$.
\end{enumerate}
\end{lem}

Moreover, we have

\begin{lem}\label{lemma logcard}
	Let $0\leq\alpha\leq p-1$. Then
	\[\lim_{n\to\infty}\frac{\log\card W([\alpha n],n)}{(\log p)n}=h(\alpha).\]
\end{lem}
\begin{proof}
	We first show that
\begin{align}\label{equality lemma logcard}
	\varliminf_{n\to\infty}\frac{\log\card W([\alpha n],n)}{(\log p)n}=h(\alpha).
\end{align}
	The proof is divided into three cases: $0\leq\alpha<(p-1)/2$, $\alpha=(p-1)/2$ and $(p-1)/2<\alpha\leq p-1$. Here, we give only the proof of the first case.

	Take $\delta>0$ such that $\alpha+\delta<(p-1)/2$. Since
	\[\card W([(\alpha+\delta)n],n)\leq\frac{n^{n\delta}}{(n\delta-1)!}\card W([\alpha n],n),\]
	by the Stirling's approximation we have
\begin{align*}
	\begin{split}
		\varliminf_{n\to\infty}\frac{\log\card W([(\alpha+\delta)n],n)}{(\log p)n}&\leq\varliminf_{n\to\infty}\frac{\log\frac{n^{n\delta}}{(n\delta-1)!}}{(\log p)n}+\varliminf_{n\to\infty}\frac{\log\card W([\alpha n],n)}{(\log p)n}\\
		&=\frac{\delta\log\frac{e}{\delta}}{\log p}+\varliminf_{n\to\infty}\frac{\log\card W([\alpha n],n)}{(\log p)n}.
	\end{split}
\end{align*}
Let $\delta\to0$ in both sides, since $\lim_{\delta\to0}(\delta\log\frac{e}{\delta})/\log p=0$, we have
\begin{align}\label{inequality log halpha}
		\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log\card W([(\alpha+\delta)n],n)}{(\log p)n}\leq\varliminf_{n\to\infty}\frac{\log\card W([\alpha n],n)}{(\log p)n}.
\end{align}
Moreover, by the properties of function $\omega(\alpha,n)$ in Lemma~\ref{theorem enumerate 2}, we have
\begin{align*}
	\begin{split}
		\card W([(\alpha+\delta)n],n)&\leq\card W\big(\big[[(\alpha-\delta)n]+1,[(\alpha+\delta)n]\big],n\big)\\
		&=h(\alpha,n,\delta)\leq([2\delta n]+1)\card W([(\alpha+\delta)n],n).
	\end{split}
\end{align*}
It follows that
\[\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log\card W([(\alpha+\delta)n],n)}{(\log p)n}=\lim_{\delta\to0}\varliminf_{n\to\infty}\frac{\log h(\alpha,n,\delta)}{(\log p)n}=h(\alpha).\]
This, together with \eqref{inequality log halpha}, yields that
\[h(\alpha)\leq\varliminf_{n\to\infty}\frac{\log\card W([\alpha n],n)}{(\log p)n}.\]
On the other hand, the inequality for the opposite direction is apparently true. So, the equality \eqref{equality lemma logcard} is established.

Since the equality $\varlimsup_{n\to\infty}\log\card W([\alpha n],n)/\big((\log p)n\big)=h(\alpha)$ can be proved as the above way, the proof of this lemma is finished now.
\end{proof}

In the sequel, we will construct a Moran set $\mathcal{W}_M(\alpha)$, where $0\leq\alpha<p-1$, to obtain the lower bound of Hausdorff dimension of $B(\alpha,\beta)$ in Theorem~\ref{theorem main theorem}. At first, we construct recursively two sequences of sets of words $\{W_n(\alpha,M)\}_{n=1}^\infty$ and $\{V_n(\alpha,M)\}_{n=1}^\infty$ below.

Let $M$ be sufficiently large such that $[\alpha M]+1<(p-1)M$. For brevity, write
\[W_1(\alpha,M)=W([\alpha M],M),\quad V_1(\alpha,M)=W([\alpha M]+1,M).\]
Suppose that the sets $W_i(\alpha,M)$ and $V_i(\alpha,M)$ are well-defined for all $1\leq i\leq n$,
then define
\begin{align*}
	\begin{split}
		W_{n+1}(\alpha,M)=\big\{&x_1\cdots x_{2^nM}\in W([\alpha 2^nM],2^nM)\colon\\
		&x_{2^{n-1}Mi+1}\cdots x_{2^{n-1}M(i+1)}\in W_n(\alpha,M)\cup V_n(\alpha,M),i=0,1\big\},
	\end{split}
\end{align*}
\begin{align*}
	\begin{split}
		V_{n+1}(\alpha,M)=\big\{&x_1\cdots x_{2^nM}\in W([\alpha 2^nM]+1,2^nM) \colon \\&x_{2^{n-1}Mi+1}\cdots x_{2^{n-1}M(i+1)}\in W_n(\alpha,M)\cup V_n(\alpha,M),i=0,1\big\}.
	\end{split}
\end{align*}
The above definitions are valid since the estimation
\begin{equation}\label{cc5}
2[\alpha2^kM]<[\alpha2^{k+1}M]+1\leq 2\left([\alpha2^kM]+1\right)
\end{equation}
holds for any $0\leq\alpha<p-1$ and $k\geq0$.

With this construction, we know that for each $n\geq1$, every word in $W_n(\alpha,M)$ is of length $2^{n-1}M$ and the sum of elements is $[\alpha2^{n-1}M]$. Similarly, every word in $V_n(\alpha,M)$ is of length $2^{n-1}M$ and the sum of elements is $[\alpha2^{n-1}M]+1$. Moreover, we even have

\begin{rem}\label{remark decompose}
	For any $0\leq i\leq n-1$, we can decompose uniquely each word in $W_n(\alpha,M)$ and $V_n(\alpha,M)$ into successive concatenations of $2^iM$-words, the sum of elements in each $2^iM$-word is $[\alpha 2^iM]$ or $[\alpha 2^iM]+1$.
\end{rem}

Based on the family of sets of $\alpha$-words $\{W_n(\alpha,M)\}_{n=1}^\infty$, define the Moran set
\begin{align*}
		\mathcal{W}_M(\alpha):=0.\prod_{n=1}^\infty W_n(\alpha,M).
\end{align*}
Then we have

\begin{lem}\label{lemma alpha M}
	Let $0\leq\alpha<p-1$, then
	\begin{align}\label{equantion alpha M}
		\lim_{M\to\infty}\dim_H\mathcal{W}_M(\alpha)=h(\alpha).
	\end{align}
\end{lem}
\begin{proof}
For the case $0\leq\alpha<(p-1)/2$, take $M$ to be large enough such that $[\alpha M]+1<[(p-1)M/2]$. By the monotonicity of function $\omega$ in \eqref{omega 3} of Lemma~\ref{theorem enumerate 2}, we have
\[\card W([\alpha M],M)=\omega(\alpha,M)\leq \omega(\alpha+1/M,M)=\card W([\alpha M]+1,M).\]
From this and the structures of words in $\mathcal{W}_M(\alpha)$ in Remark~\ref{remark decompose}, we know that
\begin{align}\label{inequality alpha M}
	\dim_H\mathcal{W}([\alpha M],M)\leq\dim_H\mathcal{W}_M(\alpha)\leq\dim_H\mathcal{W}\big(\big[[\alpha M],[\alpha M]+1\big],M\big).
\end{align}
Moreover, by Lemma \ref{lemma pqm}, we have
\[\dim_H\mathcal{W}([\alpha M],M)=\frac{\log\card W([\alpha M],M)}{(\log p)M}\]
and
\begin{align*}
	\begin{split}
		\dim_H\mathcal{W}\big(\big[[\alpha M],[\alpha M]+1\big],M\big)
		=\frac{\log\big(\card W([\alpha M],M)+\card W([\alpha M]+1,M)\big)}{(\log p)M}.
	\end{split}
\end{align*}
Thus, by Lemma~\ref{lemma logcard} we may obtain that
\[\lim_{M\to\infty}\dim_H\mathcal{W}([\alpha M],M)=\lim_{M\to\infty}\dim_H\mathcal{W}\big(\big[[\alpha M],[\alpha M]+1\big],M\big)=h(\alpha).\]
This, together with \eqref{inequality alpha M}, leads to the conclusion~\eqref{equantion alpha M}.

On the other hand, for the case $(p-1)/2\leq\alpha<p-1$, we can get similarly that
\[\dim_H\mathcal{W}([\alpha M]+1,M)\leq\dim_H\mathcal{W}_M(\alpha)\leq\dim_H\mathcal{W}\big(\big[[\alpha M],[\alpha M]+1\big],M\big)\]
for sufficiently large $M$. The proof of the remainder is similar to the first case. We omit the details here.

The proof is completed now.
\end{proof}

\section{Proof of Theorem \ref{theorem main theorem}}

In this section, we will give the proof of Theorem~\ref{theorem main theorem}.
First, we would like to present the following lemma to reveal the relations among the lower and upper digit means of $x\in I$ and the lower and upper moving digit means of $x\in I$, which will be used to achieve the upper bound of Hausdorff dimension of $B(\alpha,\beta)$.

\begin{lem}\label{lemma relation}
	For any $x\in I$, we have
	\begin{equation}
	\b{\it M}(x)\leq\b{\it A}(x)\leq\bar{A}(x)\leq\bar{M}(x).
	\end{equation}
\end{lem}
\begin{proof}
	It suffices to show that $\bar{A}(x)\leq\bar{M}(x)$. Write $\bar{M}(x)=\beta\in[0,p-1]$. Then, for any $\epsilon>0$, there exists an integer $N>0$, such that
	\[\varlimsup\limits_{m\to\infty}\frac{S_n(T^mx)}{n}<\beta+\epsilon,\quad\forall n\geq N.\]
	Furthermore, there exists an integer $Q=Q(n)\geq1$ such that
	\[\frac{S_n(T^mx)}{n}<\beta+\epsilon,\quad \mbox{i.e.},\quad \sum_{i=1}^nx_{m+i}\leq[n(\beta+\epsilon)]\]
	for any $ m\geq Q$. Suppose $t\geq Q$ and $t=Q+r+kn$, where $0\leq r\leq n-1$, then
	\[\frac{S_t(x)}{t}\leq\frac{Q+r+k[n(\beta+\epsilon)]}{Q+r+kn}.\]
	Let $t\to\infty$, then $k\to\infty$ and
	\[\varlimsup_{t\to\infty}\frac{S_t(x)}{t}\leq\frac{[n(\beta+\epsilon)]}{n}\leq\beta+\epsilon.\]
	Since $\epsilon$ is arbitrary, we have $\bar{A}(x)\leq\beta=\bar{M}(x)$.
\end{proof}

Next, we would like to present a lemma for the computation of lower bound of the Hausdorff dimension of $B(\alpha,\beta)$. Let $\mathbb{M}$ be a subset of $\mathbb{N}$. We say the set $\mathbb{M}$ is of density $\rho\in[0,1]$ in $\mathbb{N}$ if
\[\lim_{n\to\infty}\frac{\card\{i\in\mathbb{M}\colon i\leq	n\}}{n}=\rho.\]
Write
$\mathbb{N}\backslash\mathbb{M}=\{n_i\}_{i\geq1}$ where $n_i<n_{i+1}$ for all $i\geq1$. Define a
mapping $\varphi_\mathbb{M}\colon I\to I$ by
\[0.x_1x_2\ldots\mapsto 0.x_{n_1}x_{n_2}\ldots.\]
Under the mapping $\varphi_\mathbb{M}$, for any given subset $D\subset I$, we may obtain another set $\varphi_\mathbb{M}(D)=\{\varphi_\mathbb{M}(x)\colon x\in D\}$.
Moreover, we have

\begin{lem}[See Lemma 2.3 in~\cite{CT}]\label{lemma invariance}
	Suppose that the set $\mathbb{M}$ is of density zero in $\mathbb{N}$. Then for any set $D\subset I$ we have $\dim_HD=\dim_H\phi_\mathbb{M}(D)$.
\end{lem}

Lemma~\ref{lemma invariance} implies that for a set $D$, its Hausdorff dimension is invariant after deleting the digits, for which the set of their positions is of density zero in $\mathbb{N}$, from the sequences of numbers in $D$.

Now, we are ready to give the proof of Theorem~\ref{theorem main theorem}.

\begin{proof}[Proof of Theorem~\ref{theorem main theorem}]
According to Remark \ref{remark b alpha beta}, the proof is divided into three parts:
\begin{enumerate}
	    \item $\dim_HB(\alpha,\beta)=h(\beta)$ if $0\leq\alpha\leq\beta<(p-1)/2$;
		\item $\dim_HB(\alpha,\beta)=1$ if $\alpha\leq(p-1)/2\leq\beta$;
		\item $\dim_HB(\alpha,\beta)=h(\alpha)$ if $(p-1)/2<\alpha\leq\beta\leq p-1$.
\end{enumerate}
In the following, we will give the proofs of them respectively.

(1) For the upper bound, by Lemma~\ref{lemma relation}, we have $B(\alpha,\beta)\subset\bar{S}(\beta)$. It follows that $\dim_HB(\alpha,\beta)\leq\dim_H\bar{S}(\beta)=h(\beta)$.

For the lower bound, construct the set
\[\mathcal{W}_M(\alpha,\beta)=0.\prod_{n=1}^{\infty}\big(W_n(\alpha,M)\times W_n(\beta,M)^n\big).\]
Then we have: (a) $\mathcal{W}_M(\alpha,\beta)\subset B(\alpha,\beta)$; (b) $\dim_H\mathcal{W}_M(\alpha,\beta)=\dim_H\mathcal{W}_M(\beta)$.

For the proof of (a), note that for any $i\geq1$, each word in the set
\[\prod_{n=1}^i\big(W_n(\alpha,M)\times W_n(\beta,M)^n\big),\]
is of length $i2^iM$ and the length of words in $W_n(\alpha,M)$ and $W_n(\beta,M)$, $n>i$, are of common lengths $2^{n-1}M$. So,
we may decompose every number $x\in\mathcal{W}_M(\alpha,\beta)$ into successive concatenations of $2^iM$-words. Take $n$ to be sufficiently large and write $n=k2^iM+r$, $0\leq r\leq 2^iM-1$.
Then
\[\frac{(k-1)[\alpha2^iM]}{k2^iM+r}\leq\b{\it M}_n(x)\leq\frac{(k+2)([\alpha2^iM]+1)}{k2^iM+r}.\]
Let $n\to\infty$, then $k\to\infty$. It yields that
\[\frac{[\alpha2^iM]}{2^iM}\leq\b{\it M}(x)\leq\frac{[\alpha2^iM]+1}{2^iM}.\]
Since this inequality holds for all $i\geq1$, we have $\b{\it M}(x)=\alpha$ by letting $i\to\infty$. The other conclusion $\bar{M}(x)=\beta$ can be deduced in a similar manner. So, we have $\mathcal{W}_M(\alpha,\beta)\subset B(\alpha,\beta)$.

For the second assertion (b), the set of positions occupied by the words in sets $W_n(\alpha,M)$, $n\geq1$, is of density zero for any $x\in\mathcal{W}_M(\alpha,\beta)$.
By deleting all these words in the sequences of numbers in $\mathcal{W}_M(\alpha,\beta)$, we obtain the set $\mathcal{W}_M(\beta)$.
Then, (b) is established by Lemma~\ref{lemma invariance}.

By (a) and (b), we obtain that
\[\dim_HB(\alpha,\beta)\geq\dim_H\mathcal{W}_M(\alpha,\beta)=\dim_H\mathcal{W}_M(\beta).\]
Let $M\to\infty$, it yields that $\dim_HB(\alpha,\beta)\geq h(\beta)$ according to Lemma~\ref{lemma alpha M}.
The proof of this part is finished.

(2) This part is split into four cases for consideration: i) $\alpha< (p-1)/2<\beta$; ii) $\alpha=(p-1)/2<\beta$; iii) $\alpha<\beta=(p-1)/2$; iv) $\alpha=\beta=(p-1)/2$.

Case i): $\alpha< (p-1)/2<\beta$. Since $h((p-1)/2)=1$, for $\epsilon$ small enough, there exists $\delta_0>0$ and $n_0>0$ such that
\[\alpha<\frac{p-1}{2}-\delta_0<\frac{p-1}{2}+\delta_0<\beta\quad\text{and}\quad\frac{\log h\left(\frac{p-1}{2},n_0,\delta_0\right)}{(\log p)n_0}>1-\epsilon.\]
Based on the set
\[H\left(\frac{p-1}{2},n_0,\delta_0\right)=\bigg\{x_1\cdots x_{n_0}\in A^{n_0}\colon n_0(\frac{p-1}{2}-\delta_0)<\sum_{i=1}^{n_0}x_i<n_0(\frac{p-1}{2}+\delta_0)\bigg\},\]
define
\begin{align}\label{definition u n0}
	\mathcal{H}\left(\frac{p-1}{2},n_0,\delta_0\right)=0.H\left(\frac{p-1}{2},n_0,\delta_0\right)^\infty.
\end{align}
Then, by Lemma \ref{lemma pqm}, we have
\[\dim_H\mathcal{H}\left(\frac{p-1}{2},n_0,\delta_0\right)=\frac{\log\card H\left(\frac{p-1}{2},n_0,\delta_0\right)}{(\log p)n_0}=\frac{\log h\left(\frac{p-1}{2},n_0,\delta_0\right)}{(\log p)n_0}>1-\epsilon.\]
Now, construct the set
\begin{align}\label{definition w n0}
	\mathcal{W}_{M,n_0,\delta_0}(\alpha,\beta)=0.\prod_{n=1}^{\infty}\left(W_n(\alpha,M)\times H\left(\frac{p-1}{2},n_0,\delta_0\right)^{3^{n-1}}\times W_n(\beta,M)\right).
\end{align}
Similar to the proof of the foregoing part, we can also deduce that
\[\mathcal{W}_{M,n_0,\delta_0}(\alpha,\beta)\subset B(\alpha,\beta)\quad\text{and}\quad\dim_H\mathcal{W}_{M,n_0,\delta_0}(\alpha,\beta)=\dim_H\mathcal{H}\left(\frac{p-1}{2},n_0,\delta_0\right).\]
Thus, we have \[\dim_HB(\alpha,\beta)\geq\dim_H\mathcal{H}\left(\frac{p-1}{2},n_0,\delta_0\right)>1-\epsilon.\]
It proves this case since $\epsilon$ is arbitrary.

Case ii): $\alpha=(p-1)/2<\beta$. In this case, take $\delta_0$ satisfying $(p-1)/2<(p-1)/2+\delta_0<\beta$. Put
\[H'\left(\frac{p-1}{2},n_0,\delta_0\right)=\bigg\{x_1\cdots x_{n_0}\in A^{n_0}\colon \frac{p-1}{2}n_0<\sum_{i=1}^{n_0}x_i<(\frac{p-1}{2}+\delta_0)n_0\bigg\}\]
and
\[\mathcal{H}'\left(\frac{p-1}{2},n_0,\delta_0\right)=0.H'\left(\frac{p-1}{2},n_0,\delta_0\right)^\infty.\]
Then, similar to the proof of Lemma \ref{lemma logcard}, we have
\[\lim_{\delta_0\to0}\varliminf_{n_0\to\infty}\frac{\log\card H'\left(\frac{p-1}{2},n_0,\delta_0\right)}{(\log p)n_0}=\lim_{\delta_0\to0}\varliminf_{n_0\to\infty}\frac{\log\card H\left(\frac{p-1}{2},n_0,\delta_0\right)}{(\log p)n_0}=1.\]
So, for any $\epsilon>0$, there exists $n_0$ and $\delta_0$ such that \[\dim_H\mathcal{H}'\left(\frac{p-1}{2},n_0,\delta_0\right)>1-\epsilon.\]
Next, construct the Moran set $\mathcal{W}_{M,n_0,\delta_0}'\big((p-1)/2,\beta\big)$ as $\mathcal{W}_{M,n_0,\delta_0}(\alpha,\beta)$ in \eqref{definition w n0} by replacing $H\big((p-1)/2,n_0,\delta_0\big)$ with $H'\left((p-1)/2,n_0,\delta_0\right)$.
For the remaining proof of this case, it is just similar to the above discussion in Case i).

Case iii): $\alpha<\beta=(p-1)/2$. It can be proved as that of Case ii).

Case iv): $\alpha=\beta=(p-1)/2$. In this case, take $n_0$ to be even and
consider the set
\[\mathcal{W}_{n_0}\left(\frac{p-1}{2}\right)=0.W\left(\frac{p-1}{2},n_0\right)^\infty,\]
where
\[W\left(\frac{p-1}{2},n_0\right)=\bigg\{x_1\cdots x_{n_0}\in A^{n_0}\colon \sum_{i=1}^{n_0}x_i=\frac{p-1}{2}n_0\bigg\}.\]
Then we have
\[\lim_{n_0\to\infty}\dim_H\mathcal{W}_{n_0}\left(\frac{p-1}{2}\right)=h\left(\frac{p-1}{2}\right)=1\]
by Lemma \ref{lemma logcard}. It is evident that \[\mathcal{W}_{n_0}\left(\frac{p-1}{2}\right)\subset B\left(\frac{p-1}{2},\frac{p-1}{2}\right)=B\left(\frac{p-1}{2}\right).\]
Thus, $\dim_HB\big((p-1)/2\big)\geq\dim_H\mathcal{W}_{n_0}\big((p-1)/2\big)$. Let $n_0\to\infty$, then we have $\dim_HB\big((p-1)/2\big)=1$.

(3) Three cases will be considered in this part.

Case 1): $(p-1)/2<\alpha\leq\beta<p-1$. This case can be proved as that of part (1) and we omit the details here.

Case 2): $(p-1)/2<\alpha<\beta=p-1$. First, by Lemma~\ref{lemma relation}, we may obtain that $B(\alpha,p-1)\subset\b{\it E}(\alpha)$. So, $\dim_HB(\alpha,p-1)\leq\dim_H\b{\it E}(\alpha)=h(\alpha)$.

Second, construct the set
\[\mathcal{W}_M(\alpha,p-1)=0.\prod_{n=1}^{\infty}\big(W_n(\alpha,M)\times(p-1)^n\big),\]
where $(p-1)^n$ means the word $(p-1)\cdots(p-1)$ of length $n$. Then we can deduce similarly that
\[\mathcal{W}_M(\alpha,p-1)\subset B(\alpha,p-1)\quad\text{and}\quad \dim_H\mathcal{W}_M(\alpha,p-1)=\dim_H\mathcal{W}_M(\alpha).\]
Thus, $\dim_HB(\alpha,p-1)\geq\dim_H\mathcal{W}_M(\alpha)=h(\alpha)$.

The above two assertions imply that $\dim_HB(\alpha,p-1)=h(\alpha)$.

Case 3): $\alpha=\beta=p-1$. Since $B(p-1,p-1)\subset\b{\it E}(p-1)$ and $\dim_H\b{\it E}(p-1)=h(p-1)=0$, we obtain that $\dim_HB(p-1,p-1)=0=h(p-1)$.

The proof is finished now.
\end{proof}

At last, it should be pointed out that we can even study the \emph{moving digit mean} of $x$:
\begin{align}
	M(x)=\lim_{n\to\infty}\lim_{m\to\infty}\frac{S_n(T^mx)}{n},\quad x\in I.
\end{align}
Moreover, define the level sets related to it as
\begin{align}
	B^\ast(\alpha)=\left\{x\in I\colon M(x)=\alpha\right\},\quad0\leq\alpha\leq p-1.
\end{align}
In this situation, the set $B^\ast(\alpha)$ is somewhat trivial because we have
\begin{thm}
	Let $0\leq\alpha\leq p-1$. If $\alpha=0,1,\ldots,p-1$, then $B^\ast(\alpha)$ is a countable set. Otherwise, $B^\ast(\alpha)$ is an empty set. Hence, we always have
	\begin{align}
		\dim_HB^\ast(\alpha)=0
	\end{align}
	for any $0\leq\alpha\leq p-1$.
\end{thm}
\begin{proof}
  Let $\alpha=i$, where $i=0,1,\ldots, p-1$. Then each number in $B^\ast(\alpha)$ is ultimately 1-periodic ending with $i^\infty$, that is
  \[B^\ast(\alpha)=\{x\in I\colon x=0.x_1x_2\ldots x_niii\ldots,n\geq 1\}.\]
  Thus, $B^\ast(\alpha)$ is countable. On the other hand, if $i<\alpha<i+1$, where $i=0,1,\ldots$, or $p-2$, then for any $x\in B^\ast(\alpha)$ and $n\geq1$ the limit $\lim_{m\to\infty}S_n(T^mx)/n$ does not exist according to the proof by contradiction and Cauchy's criterion for convergence of sequences. It follows that $B^\ast(\alpha)=\emptyset$.
\end{proof}

\subsection*{Acknowledgment}
This work was finished when the author visited the Laboratoire d'Analyse et de Math\'{e}matiques Appliqu\'{e}es, Universit\'{e} Paris-Est Cr\'{e}teil Val de Marne, France. Thanks a lot for the great encouragement from his collaborator and the assistance provided by the laboratory.

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2018 – a year of achievement for Aurivo Co-Op
In a year marked by ongoing volatility in global dairy markets, which in turn impacted farmers right across the country, 2018 was a year of overall achievement and progress for Aurivo – the globally-focused Agribusiness, headquartered in the North West of Ireland.
Commenting, Aaron Forde, CEO of Aurivo said: "By and large 2018 was a challenging year for the Irish dairy sector – from volatile global markets to adverse weather conditions in Ireland. Against that backdrop however, with an eye to the future, Aurivo strove hard to ensure that it was a year of achievement for the Co-Op. As an innovative multi-purpose Co-Operative, we have a strong ambition to continue to grow a sustainable business that will not only create value for our members but ensure a certain future for our farms, our processes and our communities for generations to come. We look forward to an even more positive year ahead as we continue to expand our business and effectively continue to be one of the strongest member-owned organisations in the country."
Key achievements in 2018:
Strategy and investment
– Aurivo's focus and achievements in 2018 were grounded by the Co-Op's Engage 2022 strategic growth plan, which in turn is underpinned by the Group's €48 million five-year capital investment programme that was announced in 2018. The ongoing investment programme, designed to enable innovative new product development and to further position the business to meet global demand and anticipated market growth in the years ahead, incorporates a range of projects across the Co-Op including the extensive €26m upgrade of the Diary Ingredients facility in Ballaghaderreen Co. Roscommon.
– Working to deliver on the goals of Engage 2022, Aurivo saw changes to its leadership on its Executive Team with the senior appointments of Sheila Maloney to General Manager of Aurivo's Agribusiness and Marilyn Phillips to Group Head of HR. Stephen Blewitt in turn was appointed to lead Aurivo's Dairy Ingredients and Consumer Foods business.
Supporting farmers
– To assist our suppliers with the challenges they were facing in 2018, Aurivo launched the innovative MilkFlex Loan Scheme which offered dairy farmers, flexible, competitively-priced financing.
– With price volatility being a feature of 2017 also, it was gratifying to see that within the KPMG / Irish Farmers Journal Annual Milk Price Review announced last July that Aurivo has lead its neighbouring Co-Ops for four of the past five years on milk price.
– A good example of Aurivo's ongoing produce development and innovation was the launch in September of Connacht Gold MÓR Milk, a fresh and nutrient-rich whole milk specifically formulated for children from 1-12 years.
– Aurivo's achievements were recognised by a range of awards in 2018:
Aurivo Consumer Foods was named Best Dairy Supplier at the Checkout 'Best in Fresh' awards, before scooping the prize for Best Chill & Fresh Supplier at the Barry Group Awards in September.
In early October, there were four wins at the Blas na hÉireann awards. Aurivo's Connacht Gold Creamery Butter was awarded silver in the Dairy category, while its Connacht Gold Softer Butter took bronze in the same grouping. In the Flavoured Milk category, the Co-op won silver for Dunnes Stores Chocolate Flavoured Milk and bronze for Dunnes Stores Strawberry Flavoured Milk. Dunnes Stores uses milk supplied by Aurivo to develop its own brand products.
The Mums and Tots Awards, also in October, saw Connacht Gold MÓR Milk being named Best Follow on Milk.
Aurivo Consumer Foods was also named Supplier of the Year at the Tesco Fresh Supplier Awards.
Corporate responsibility:
– Aurivo's Charity of the Year programme is an important part of its Corporate Social Responsibility and Origin Green sustainability strategy. During 2018 Aurivo employees fundraised for Down Syndrome Ireland's local branches across the North West region as part of the Co-Op's Charity of the Year programme.
– For the second year Aurivo coordinated the Gaffney and Mulleady Agriculture, Food and Nutrition Scholarship which was established in memory of Paddy Gaffney and Sean Mulleady who died tragically in 2013.
Commenting on 2018, Pat Duffy, Chairman of Aurivo said: "This is a confident business delivering for its members, and it is within that context that we highlight some of our key achievements in 2018. Aurivo has a long track record of achievement, much of this pointed out in the recent publication of 'Fields of Gold' – one of the most comprehensive studies ever undertaken of farming west of the Shannon. While we are proud of our history and track record, our focus is firmly on the future as we strive to continuously achieve and deliver for our members, our farmers, our employees and the communities we operate in."
Aurivo's financial and operational performance for 2018 will be announced in April.

redpajama_set_name: RedPajamaCommonCrawl


text:
O Tratado das Nove Potências (em japonês:九 カ国 条約, Kyūkakoku Jōyaku) ou Acordo das Nove Potências (em chinês: 九 国 公约) foi um tratado afirmando a soberania e a integridade territorial da China, de acordo com a Política de portas abertas, assinada por todos os participantes da Conferência Naval de Washington em 6 de fevereiro de 1922.

O secretário de Estado dos Estados Unidos, John Hay tinha emitido a sua "notas sobre as portas abertas", de Setembro a Novembro de 1899, seguido por uma circular diplomática em julho de 1900, pedindo que todas as grandes potências mundiais com interesses na China declarassem formalmente que iriam manter uma política de "portas abertas" para permitir que todas as nações, a igualdade de direitos e igualdade de acesso aos Tratados Portos dentro de suas esferas de influência na China. Temendo que as potências européias e o Japão estivessem se preparando para repartir a China em colônias, Hay também acrescentou disposições que a integridade territorial e administrativa chinesa deveria ser mantida.

Embora nenhum país especificamente respaldou a proposta de Hay, este anunciou que cada uma das potências tinham autorizado de princípio, e que os tratados feitos a partir de 1900 faziam referência à política de portas abertas. No entanto, a concorrência entre as diversas potências para concessões especiais dentro da China (pelos direitos de estrada de ferro, os direitos de mineração, os empréstimos, os portos ao comércio estrangeiro, etc) continuaram inabaláveis.

Os Estados Unidos estavam particularmente desconfiados dos planos japoneses na China após a Guerra Russo-Japonesa (1904-1905) e das Vinte e Uma Exigências (1915) e repetidamente assinaram acordos com o governo japonês se comprometendo a manter uma política de igualdade na Manchúria e na China. Estes acordos culminaram com o Acordo Lansing-Ishii em 1917, que logo foi mostrado ser completamente ineficaz.

Durante a Conferência Naval de Washington de 1921-1922, o governo dos Estados Unidos voltou a levantar a Política de Portas Abertas como uma questão internacional, e apelou a todos os participantes (Estados Unidos, Japão, China, França, Grã-Bretanha, Itália, Bélgica, Holanda, e Portugal), que assinassem um novo tratado que visa tornar a Política de Portas Abertas parte da lei da política internacional.

No entanto, o Tratado das Nove Potências carecia de qualquer regulamentação executiva e, quando foi violado pelo Japão durante a invasão da Manchúria em 1931 e na criação de Manchukuo, os Estados Unidos não puderam fazer muito mais do que protestar e impor sanções econômicas.

Assim, a Segunda Guerra Mundial efetivamente anulou o Tratado das Nove Potências.

O Tratado das Nove Potências foi um dos vários tratados celebrados na Conferência Naval de Washington. Outros acordos importantes incluíram o Tratado das Quatro Potências e o Tratado das Cinco Potências.

Bibliografia

Ligações externas

Tratados dos Estados Unidos
Tratados do Império do Japão
Tratados assinados em 1922
História da China

redpajama_set_name: RedPajamaWikipedia


I'm bothered by the fact that to make great work Lubalin was forced to have these embarassing sedimentary layers underneath. I'm bothered by his conviction that this is the only way that an agency can be run. I'm bothered by the suggestion that no-one will pay for great work, or that this sort of product cannot be exchanged with a respectable sum in return.
There's another problem too; it's heavily implied in the retelling of this story that one cannot gain independence from those first two floors and that we'll be shackled to them forever, regardless of our profession or the nature of our limited success.
For some reason the description of Lubalin's office has me drawing parallels with this piece by Clay Shirky on how the internal culture of a company has such drastic and unforeseen consequences on their products. And I'm clearly fumbling for connections between the two articles here but burueacracy, whether enforced by government protocols, or the natural strains of the marketplace, is kinda sucky either way.
I wonder what the people on the first two floors thought of people on the third.


This month, these parishes covering Burlington, Mercer, Monmouth and Ocean Counties, are getting a clearer idea of what that new course might look like following the issuance of preliminary recommendations by the Diocesan Planning Commission.
With its focus on consultation that includes parishioners, lay leaders, clergy and diocesan leaders, Faith in Our Future is an 8-step process that will culminate with the July 2017 implementation of decisions made by Bishop David M. O'Connell, C.M. The process has now just entered Steps 4 and 5, after preliminary recommendations were developed by the Commission during a 3-day retreat in mid-June.
Those recommendations were sent to the 25 Cohorts (a grouping of four to six parishes based on proximity to one another and other relevant factors) the week of June 27 for further discussion and response.
The Planning Commission will meet again in October to review the Cohort responses and bring together a set of final recommendations for Bishop O'Connell to consider. This will lead to the last two steps — consultation and decision-making by the Bishop (expected to take place by the end of 2016 or the beginning of 2017) and preparation for implementation (to begin early in 2017). Implementation of the Bishop's decisions is expected to begin July 1, 2017.
The Planning Commission's recommendations focused on four of the six possible Parish Models for each community within the Cohort: 1) Collaboration; 2) Linking; 3) Merger, and 4) "In Solidum" team.
The remaining models that have not been among the preliminary recommendations are: 5) Parish Life Director, and 6) Personal Parish.
In customized reports written to each Cohort, the Planning Commission expressed appreciation for the efforts of Parish Core Team members and Cohort representatives, commending each for the quality, depth and sense of mission reflected in the work. The Commission also acknowledged the different suggestions and rationales submitted by the Cohorts, pointing out areas of agreement and disagreement between them and the Commission.
In a few cases, the Commission cited Cohorts that submitted incomplete information or suggestions that were deemed unrealistic and/or unsupported by data. Some Cohorts were asked to submit the missing information when they file their responses by October.
In rendering their preliminary recommendations, the Commission took into account the Cohorts' suggestions; the data and input on which they were based, such as parish size, debt and findings in parish surveys, and the different makeup of each Cohort, such as the ethnic breakdown, the presence of Catholic schools and hospital ministries that must be supported, and the existence of a religious order ministering in one or more parishes.
The recommendations also went beyond the six Parish Models, finding different and specific ways for parishes to work together, particularly as Cohorts, even after possible mergers or other restructuring. For instance, the recommendations promoted the idea for certain Cohorts to develop an identity as a regional faith community; to hire a professional business administrator or youth minister to serve all parishes within a Cohort; to coordinate schedules to reduce the number of Masses and increase attendance at each across the Cohort, and to study ways that buildings can serve the Cohort, as opposed to just the parish.
In letters sent June 28 to pastors and to Parish Core Team members, Msgr. Leonard F. Troiano, episcopal vicar for planning, encouraged ongoing consultation over the next several months within parish communities to fully digest what was being asked and consider how they might respond.
He expects that with parish input the Cohorts will be able to provide a substantive response back to the Planning Commission that could affirm one or more of the preliminary recommendations or suggest a different course of action.
To view the recommendations, go to dioceseoftrenton.org/faithinourfuture.


From left: President of AMCHAM, Zulfikar Ally, US Ambassador to Guyana, Sarah-Ann Lynch and Minister of Foreign Affairs, Carl Greenidge.
across all sectors," says newly-accredited US Ambassador to Guyana, Sarah-Ann Lynch.
She was speaking at a welcome reception in her honour hosted by the American Chamber of Commerce Guyana (AMCHAM), on Tuesday evening.
The event was also to formally launch the AMCHAM's office in the Guyana Marriott Hotel.
Given the increased relevance of Guyana in the international community, and attraction of the country as one of the preferred destinations for investments for American businesses over the last seven to eight years, President of AMCHAM, Zulfikar Ally explained that since the launch of the Chamber in August 2018, they already have 45 active members with 7 committees.
Ally noted that their members can participate and contribute to the development of the organisation, share ideas for economic growth between Guyana and the US, and express and brainstorm solutions of concerns affecting their business.
He welcomed Ambassador Lynch to Guyana and noted that the US envoy has come to Guyana at a convenient time when the increased business activity between the two countries has led to the creation of new relationships and strengthening of old ones.


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Watch trailer for Malala Yousafzai documentary 'He named me Malala'
20 Jun, 2015 | 6:38 pm
Written by Staff Writer 20 Jun, 2015	| 6:38 pm
Fox Searchlight has released a trailer for "He Named Me Malala," an upcoming documentary feature film on the life of 2014 Nobel Peace Prize recipient Malala Yousafzai.
The film, directed by Oscar-winning filmmaker Davis Guggenheim, gives an insider's portrait of Yousafzai, who has become a global champion of education rights for women.
The documentary centers on Nobel Peace Prize Laureate Malala Yousafzai, who was targeted by the Taliban and severely wounded by a gunshot when returning home on her school bus in Pakistan's Swat Valley.
Since then, Malala has become a symbol of defiance in the fight against militants operating in ethnic Pashtun areas in northwest Pakistan.
She has met various world leaders, including U.S. president Barack Obama and Britain's Queen Elizabeth, and was lucky enough to address the United Nations, and at the age of 17 became the youngest person in history to receive the Nobel Peace Prize, in 2014.
"He Named Me Malala" is being released by Fox Searchlight in association with Image Nation Abu Dhabi and Participant Media will be released later this year.
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PICAYUNE — A company building a sand-drying plant to service the oil industry is eying an August start-up.
Jacque Ballay, vice president in charge of marketing and sales for Alliance Consulting Group, LLC, tells the Picayune Item when the plant cranks up, it will be able to handle a maximum of 200 truckloads of sand per day, based on demand for the product.
The plant covers 30 acres and stretches along the Norfolk-Southern railway from Picayune's southern city limits to Holcomb's Crossing near Nicholson community.
Ballay said 50 workers would run the operation. He says the plant might initiate another round of hiring based on the demands of the market, when the company begins production.
"Depending on what our orders are, we have plans for expanding the facility, and that would entail hiring more employees," said Ballay.
He said phase two of the plant calls for installing another drier, he said.
The plant will process the sand, which will be mined from local and area pits, producing a fine sand for the oil drilling industry. Ballay said it would be shipped throughout the United States by truck and rail.
Alliance Consulting Group is headquartered in New Orleans.


By now, the whole world has seen the host of appalling videos concerning the gruesome practices of Planned Parenthood. This poem is my response to the scandal. Since the subject matter is immensely important, I felt that brevity was out of the question: this poem had to be in the form of a heroic crown of sonnets (which I have previously done twice, once for St. Valentine's Day and the other for the Way of the Cross).
While no written word can ever do complete justice to gravity of this topic, I pray that what follows at least conveys, in a measured way, the reasons for the anger, consternation, sadness, and utter horror expressed by many people of good will. By explicitly making a parallel to the plight of the Holy Innocents, perhaps I also wished to find some spark of glory amidst the tragedy, appealing to Christian hope and God's mercy that, like the sons of Bethlehem, today's victims of abortion, who perish without the grace of baptism, might also enjoy the beatific vision.
against the power of a sharpened knife.
and unto captives, liberty proclaim?
and ransom man from his enduring shame?
Herodian blood-thirst and wild deceit.
and tilled by the most wicked hands of Cain.
flesh flushed away to some unworthy tomb.
or shall we, held by fear, fail to defend?
to strut and fret in vain for but an hour?
low, helpless lives consumed by the elite.
demands of those demanding high supply.
for limbs and organs of a plundered child.
her sacrifice unto the god of Choice.
you barely hear the cry of Rachel's young.
no plaintive tune gives grief an apt disguise.
unto oblivion with chilling ease.
"per me si va…" is written on her thighs.
in deepest valleys, closest to the heart.
and blot the colored promise in the sky.
death's flood runs red with unarrested thrill.
no mourning bells for them are ever rung.
a harmony which only sin completes.
to praise the perpetrators of the lie.
no priest can find a blessing to impart.
the justice scales that judges overthrew.
but leaves them fallen on the icy field.
oblations to a strange god ever rise.
another poor and banished child of Eve.
and Isaac's angel saves none from the grave.
like voodoo knives into a lifeless doll.
cloaked only in their dazzling scarlet streams.
who writhe in agony to no avail.
exchanged for what the publican supplies.
a multitude of youths bereft of voice.
no diff'rent from the price of wooden beams.
in Italy can buy the serpent's hiss.
their tragedy unravels in our sight.
of children and the massacre shall cease.
to spare these children from man's deathly rage.
shall they perceive celestial delight?
and know for all eternity its toll?
that theirs was not a futile holocaust.
that they will not have been conceived in vain.
O, would that we might live to see at end!
Low, helpless lives consumed by the elite?


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(SPOT.ph) A Philippine Air Force (PAF) pilot was killed in a plane crash in Bagac, Bataan yesterday while another is still missing, according to ABS-CBN News, citing a newsreport on Bandila.
Two pilots were flying one of the PAF's S-211 jet trainer plane when it crashed at 3 p.m. in Bagac. The PAF have not disclosed the identity of the dead pilot whose body was beyond recognition, according to the Bandila report. The second pilot on the plane, Capt. Raymond de Leon, is still missing.
According to PAF spokesperson Lt. Col. Miguel Ocol, the two pilots went on a proficiency flight to add to their flying hours. The Philippine Coast Guard did a "surface search" yesterday to search for remaining debris from the crash. According to Bandila, the Coast Guard will launch a full-fledged search and retrieval operation today, April 29.


Funding: Inspection and enforcement agencies have varied levels of budgets and personnel to finish inspections and enforcement. We will additionally make it easier to in selecting a compounding pharmacist for your business you probably have that want. It isn't as generally practiced as it had been however it is still a viable possibility should your pharmacy supply it. compounding pharmacy can convert these pills into a liquid for consumption.
I am beginning my 2nd yr in Pharmacy College (Year 4/6 whole) and I knew I needed to turn into a Pharmacist since Junior 12 months in high school. Effectively…what I know now…that I did not know then…are things not essentially associated to pharmacy. He began his personal private observe right here at the Lime within the Coconut Specialty Compounding Pharmacy when he moved again to Missouri in 2011.
David G. Miller, RPh, govt vice president and CEO, International Academy of Compounding Pharmacists. He's the co-founder and former head pharmacist of New England Compounding Middle in Framingham. But growing public backlash against that value has resulted within the entry of new rivals – compounding pharmacies capable of produce comparable versions for a fraction of the price.
Compounding for acne is an excellent approach to get precisely what your skin wants in the most handy manner for you attainable. The Drug Enforcement Administration regulates compounding pharmacies' handling of controlled substances. We're licensed by the California State Board of Pharmacy to offer sterile formulations.
PCAB® was created by the main organizations within the pharmacy career for the purpose of recognizing these compounding pharmacies that have demonstrated their outstanding commitment to quality. I think these cheaper merchandise are being produced by what is called bulk compounding from uncooked materials.
If i cant work as a pharmacy tech i would positively do neighborhood service in any local pharmacy if i have to. The IVG Veterinary Compounding Pharmacy is the one sterile and non-sterile pharmacy dedicated to veterinary-only compounding in New England. And in keeping with their website, the credits switch to pharmacy colleges within the state so…IDK.
Our state-of-the-art compounding services contain the newest applied sciences to give you the very best high quality compounds. Nevertheless, the elaboration of such compounds remains pharmacy technician certification study guide 2017 individualized: it's performed on a one-by-one foundation and the compounding pharmacy must file a copy of the prescription for each compound.
I began to work as a tech just about out of straightforward curiosity, combined with a job opening in the pharmacy dept. However I do imagine pharmacy career options are pretty various, and can be so for quite a while. Traditional pharmacies will proceed to be regulated by state boards of pharmacy.
The law signed in November aims to make clear the FDA's authority to regulate compounding pharmacies after federal regulators have been blamed for not appearing soon enough to close down the pharmacy at the center of the meningitis outbreak. As more households became linked to the Internet, client on-line providers reminiscent of Prodigy, America Online and Compuserve grew to become more and more standard.
There are a lot of areas that may involve compounding to make the taking of medicines simpler and more practical. Back in our compounding lab, we frequently get calls from our veterinarian prescribers on the lookout for flavored low dose liquid aspirin for pets with sure cardiovascular conditions (identical to grownup people have).
This was really good 🙂 I am 31 years old and worked as a pharmacy tech once I was A LOT youthful. Compounding provides GPs with a broader vary of remedy choices: many more dosage codecs and dose flexibility. Neighborhood service inside an precise pharmacy is unlikely, attributable to privacy laws, and many others.
If you would like to be taught more about compounding medication and the way it may be able to profit your particular scenario, stop by our pharmacy location or call us at 573-964-6786. Downside is…I reside in Florida the place it's essential to get certification to even think about turning into a pharmacy technician.
I've another presentation on compounding basically that I've offered to doctor groups, nursing groups, pharmacy teams, hospice organizations, and to the general public as properly. As a former preceptor with numerous contact with college students…I've seen how robust it can be. But if you happen to get accepted to pharmacy school…it's as a result of you could have proven the abilities needed to succeed.
The fact that Virginia obtained the medication from a compounding pharmacy magnifies the risk of the deadly injection being painful and ineffective, Grey's attorneys argue. We're a full service compounding pharmacy, and have made just about pharmacy technician book all the things within the many years we have now been round.
The Texas case is 4:13-cv-02901 in the U.S. District Court docket for the Southern District of Texas Houston Division. Compounding doesn't discriminate; athletes from all completely different disciplines have enjoyed great results by taking compounded medicine. A twin degree will assist if you wish to move up in administration with pharmacy or within the pharma world.
It is sad that chains, mail order and deep discounters have used pharmacy to draw in sufferers, to increase sales of OTC products. I believe pharmacy is great occupation one you don't have to deal with fluid/wounds and two you're type like your individual boss.


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'SNL' captures the critical moment before Kendall Jenner's Pepsi ad shoot
"And then Kendall Jenner comes up to a police officer and gives him a Pepsi, everybody celebrates, and ... no? Are you sure that's tone-deaf?"
Yeah. We're sure. We're all very sure.
SEE ALSO: Pepsi pulls cringeworthy Kendall Jenner ad after massive backlash
In a digital short at the top of Saturday Night Live, Beck Bennett plays the one person — the ONLY person — who was clearly not so sure that Kendall Jenner's Pepsi commercial was going to be a total catastrophe.
Bennett (below) is ostensibly the director of the absurdly, jaw-droppingly off-target ad that came late last week. In the clip, he speaks with a series of people on his cell phone just minutes before it's time to shoot.
Are you ... sure it's tone-deaf?
"No, we're celebrating the cultures — everybody celebrates!" he pleads.
But it doesn't matter how he tries to justify it, whoever's on the other line (see: ALL OF US) won't give him the vote of confidence he's after.
"Oh. Got it. Just kind of using them to sell soda? Got it."
Whoever he speaks with, it's clear: This idea is not going to fly.
"Hey could you maybe put a neighbor on the phone? A black one?" he asks.
And once again, not getting what he wants.
"So the whole thing is kind of an homage to the Black Lives Matter movement ... don't even touch it?"
Problem is, the shoot is about to begin; cameras are literally rolling.
This trainwreck is too far down the tracks to stop.
And here comes the talent.
Train, wrecked.
"I stop the police from shooting black people by handing them a Pepsi.
"I know, it's cute, right?"
WATCH: Stephen Colbert and 'SNL' have figured out the right way to use President Trump
Topics: Entertainment, kendall jenner, kendall-jenner-pepsi-commercial, Pepsi, Saturday Night Live, Television