Arithmetic representations of real numbers in terms of self-similar sets

Abstract: Suppose $n\geq 2$ and $\mathcal{A}{i}\subset {0,1,\cdots ,(n-1)}$ for $ i=1,\cdots ,l,$ let $K{i}=\bigcup\nolimits_{a\in \mathcal{A}{i}}n^{-1}(K{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in \mathbb{Z}$ with $\prod\nolimits_{i}m_{i}\neq 0,$ we let \begin{equation*} S_{x}=\left{ \mathbf{(}y_{1},\cdots ,y_{l}\mathbf{)}:m_{1}y_{1}+\cdots +m_{l}y_{l}=x\text{ with }y_{i}\in K_{i}\text{ }\forall i\right} . \end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \begin{equation*} U_{r}={x:\mathbf{Card}(S_{x})=r}, \end{equation*} where $\mathbf{Card}(S_{x})$ denotes the cardinality of $S_{x}$, and $r\in \mathbb{N}^{+}$. We prove under the so-called covering condition that the Hausdorff dimension of $U_{1}$ can be calculated in terms of some matrix. Moreover, if $r\geq 2$, we also give some sufficient conditions such that the Hausdorff dimension of $U_{r}$ takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of $U_{r}$ is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.