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

Abstract

Suppose $n\ge 2$ and ${\mathcal{A}}_i\subset \{0,1,\cdots ,(n-1)\}$ for $i=1,\cdots ,l,$ let $K_i={\bigcup }_{a\in {\mathcal{A}}_i}{n}^{-1}(K_i+a)$ be self-similar sets contained in $[0,1].$ Given $m_1,\cdots ,m_l\in \mathbb{Z}$ with ${\prod }_i{m}_i\ne 0,$ 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 $r\ge 2$, 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.

19 pages