Taiwanese Journal of Mathematics
Taiwanese J. Math.
WEIGHTED REPRESENTATION FUNCTIONS ON $\mathbb{Z}_m$
Quan-Hui Yang and Yong-Gao Chen
More by Quan-Hui Yang
More by Yong-Gao Chen
Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For $A\subseteq \mathbb{Z}_m$ and $n \in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\hat{r}_{k_1,k_2}(\mathbb{Z}_m \setminus A,n) = \hat{r}_{k_1,k_2}(A,n)$ for all $n\in \mathbb{Z}_m$. As a corollary, we prove that there exists $A\subseteq \mathbb{Z}_m$ such that $\hat{r}_{k_1,k_2}(\mathbb{Z}_m\setminus A,n)=\hat{r}_{k_1,k_2}(A,n)$ for all $n\in \mathbb{Z}_m$ if and only if $2d \mid m$, where $d=(k_1,m)(k_2,m)/(k_1,k_2,m)^2$. We also pose several problems for further research.
Taiwanese J. Math., Volume 17, Number 4 (2013), 1311-1319.
First available in Project Euclid: 10 July 2017
https://projecteuclid.org/euclid.twjm/1499706119
doi:10.11650/tjm.17.2013.2463
Primary: 11B34: Representation functions 11L03: Trigonometric and exponential sums, general
representation function partition Sárközy problem
Yang, Quan-Hui; Chen, Yong-Gao. WEIGHTED REPRESENTATION FUNCTIONS ON $\mathbb{Z}_m$. Taiwanese J. Math. 17 (2013), no. 4, 1311--1319. doi:10.11650/tjm.17.2013.2463. https://projecteuclid.org/euclid.twjm/1499706119
R. Balasubramanian and G. Prakash, On an additive representation function, J. Number Theory, 104 (2004), 327-334.
Y.-G. Chen, On the values of representation functions, Sci. China Math., 54 (2011), 1317-1331.
Y.-G. Chen and M. Tang, Partitions of natural numbers with the same representation functions, J. Number Theory, 129 (2009), 2689-2695.
Y.-G. Chen and B. Wang, On additive properties of two special sequences, Acta Arith., 110 (2003), 299-303.
J. Cilleruelo and J. Rué, On a question of Sárközy and Sós for bilinear forms, Bull. Lond. Math. Soc., 41 (2009), 274-280.
G. Dombi, Additive properties of certain sets, Acta Arith., 103 (2002), 137-146.
Digital Object Identifier: doi:10.4064/aa103-2-3
A. Dubickas, A basis of finite and infinite sets with small representation function, Electron. J. Combin., 19 (2012), R6.
P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, I, Pacific J. Math., 118 (1985), 347-357.
P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, II, Acta Math. Hungar., 48 (1986), 201-211.
P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, III, Studia Sci. Math. Hungar., 22 (1987), 53-63.
P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, IV, in: Number Theory, Proceedings, Ootacamund, India, 1984, Lecture Notes in Math., Vol. 1122, Springer-Verlag, Berlin, 1985, pp. 85-104.
P. Erdős, A. Sárközy and V. T. Sós, Problems and results on additive properties of general sequences, V, Monatsh. Math., 102 (1986), 183-197.
P. Erdős and P. Turán, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc., 16 (1941), 212-215.
G. Grekos, L. Haddad, C. Helou and J. Pihko, Representation functions, Sidon sets and bases, Acta Arith., 130 (2007), 149-156.
G. Horváth, On additive representation function of general sequences, Acta Math. Hungar., 115 (2007), 169-175.
S. Z. Kiss, On the $k$-th difference of an additive representation function, Studia Sci. Math. Hungar., 48 (2011), 93-103.
V. F. Lev, Reconstructing integer sets from their representation functions, Electron. J. Combin., 11 (2004), R78.
M. B. Nathanson, Representation functions of sequences in additive number theory, Proc. Amer. Math. Soc., 72 (1978), 16-20.
M. B. Nathanson, Representation functions of additive bases for abelian semigroups, Int. J. Math. Math. Sci., 2004(30) (2004), 1589-1597.
M. B. Nathanson, Inverse problems for representation functions in additive number theory, Surveys in number theory, 89-117, Dev. Math., 17, Springer, New York, 2008.
J. L. Nicolas, I. Z. Ruzsa and A. Sárközy, On the parity of additive representation functions, J. Number Theory, 73 (1998), 292-317.
C. Sándor, Partitions of natural numbers and their representation functions, Integers, 4 (2004), A18.
A. Sárközy and V. T. Sós, On additive representation functions, in: The Mathematics of Paul Erdős, (R. L. Graham and J. Ne\ptmrs šet\ptmrs řil, eds.), Springer-Verlag, 1997, pp. 129-150.
M. Tang, Partitions of the set of natural numbers and their representation functions, Discrete Math., 308 (2008), 2614-2616.
Digital Object Identifier: doi:10.1016/j.disc.2007.06.006
Q.-H. Yang and F.-J. Chen, Partitions of $\mathbb{Z}_m$ with the same representation functions, Australas. J. Combin., 53 (2012), 257-262.
Q.-H. Yang and Y.-G. Chen, Partitions of natural numbers with the same weighted representation functions, J. Number Theory, 132 (2012), 3047-3055.
Mathematical Society of the Republic of China
Taiwanese Journal of Mathematics Home
A REMARK ON WEIGHTED REPRESENTATION FUNCTIONS
Qu, Zhenhua, Taiwanese Journal of Mathematics, 2014
The supremum of autoconvolutions, with applications to additive number theory
Martin, Greg and O'Bryant, Kevin, Illinois Journal of Mathematics, 2009
Rumely's local global principle for Weakly P$\calS$C Fields over Holomorphy Domains
Razon, Aharon and Jarden, Moshe, Functiones et Approximatio Commentarii Mathematici, 2008
On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors
Khatri, C. G., The Annals of Mathematical Statistics, 1966
John functions, quadratic integral forms and o-minimal structures
Kurdyka, K. and Xiao, J., Illinois Journal of Mathematics, 2002
On A. C. Limits of Decreasing Sequences of Continuous or Right Continuous Functions
Grande, Zbigniew, Real Analysis Exchange, 2000
Some Distribution Problems in the Multivariate Complex Gaussian Case
Pillai, K. C. S. and Jouris, G. M., The Annals of Mathematical Statistics, 1971
Uniform Approximation by Bivariate Step Functions Quasicontinuous with Respect to Single Coordinates
Richter, Christian, Real Analysis Exchange, 2008
On Schauder basis properties of multiply generated Gabor systems
Nielsen, Morten, Rocky Mountain Journal of Mathematics, 2016
Asymptotic profile of blow-up solutions of Keller-Segel systems in super-critical cases
Sugiyama, Yoshie, Differential and Integral Equations, 2010
euclid.twjm/1499706119