  In this paper we consider the relationship between the Assouad and
box-counting dimension and how both behave under the operation of taking
products. We introduce the notion of `equi-homogeneity' of a set, which
requires a uniformity in the size of local covers at all lengths and at all
points. We prove that the Assouad and box-counting dimensions coincide for sets
that have equal upper and lower box-counting dimensions provided that the set
`attains' these dimensions (analogous to `s-sets' when considering the
Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show
that for any $\alpha\in(0,1)$ and any $\beta,\gamma\in(0,1)$ such that $\beta +
\gamma\geq 1$ we can construct two generalised Cantor sets $C$ and $D$ such
that $\text{dim}_{B}C=\alpha\beta$, $\text{dim}_{B}D=\alpha\gamma$, and
$\text{dim}_{A}C=\text{dim}_{A}D=\text{dim}_{A}(C\times
D)=\text{dim}_{B}(C\times D)=\alpha$.

Classical Analysis and ODEs