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

Suppose $n\ge 2$ 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)$beself-similarsetscontainedin$[0,1].$Given$ m_{1},\cdots ,m_{l}\in $\mathbb{Z}$$with$\prod\nolimits_{i}m_{i}\neq 0,$ we let \begin{equation*} S_{x}=\left{ $($y_{1},\cdots ,y_{l}$)$:m_{1}y_{1}+\cdots +m_{l}y_{l}=x$\text{ with }$y_{i}\in K_{i}$$\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:$\mathbf{C}\mathbf{a}\mathbf{r}\mathbf{d}$(S_{x})=r}, \end{equation*} where $\mathbf{C}\mathbf{a}\mathbf{r}\mathbf{d}(S_x)$ 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 $r\ge 2$, 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.