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.
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