On the distribution of the Cantor-integers

Abstract

For any positive integer $p\geq 3$, let $A$ be a proper subset of $\{0,1,\ldots, p-1\}$ with $\sharp A=s\geq 2$. Suppose $h: \{0,1,\ldots,s-1\}\to A$ is a one-to-one map which is strictly increasing with $A=\{h(0),h(1),\ldots,h(s-1)\}$. We focus on so-called Cantor-integers $\{a_n\}_{n\geq 1}$, which consist of these positive integers $n$ such that all the digits in the $p$-ary expansion of $n$ belong to $A$. Let $\mathfrak{C}=\left\{\sum\limits_{n\geq 1}\frac{\varepsilon_n}{p^n}: \varepsilon_n\in A \text{ for any positive integer } n\right\}$ be the appropriate Cantor set, and denote the classic self-similar measure supported on $\mathfrak{C}$ by $\mu_{\mathfrak{C}}$. Now that $n^{\log_s p}$ is the growth order of $a_n$ and $\left\{\frac{a_n}{n^{\log_s p}}:~n\geq 1\right\}'$ is precisely the set $\left\{\frac{x}{(\mu_{\mathfrak{C}}([0,x]))^{\log_s p}}: x\in\mathfrak{C}\cap[\frac{h(1)}{p},1]\right\}$, where $E'$ is the set of limit points of $E$, we show that $\left\{\frac{a_n}{n^{\log_s p}}:~n\geq 1\right\}'$ is just an interval $[m,M]$ with $m:=\inf\left\{\frac{a_n}{n^{\log_s p}}:n\geq 1\right\}$ and $M:=\sup\left\{\frac{a_n}{n^{\log_s p}}:n\geq 1\right\}$. In particular, $\left\{\frac{x}{(\mu_{\mathfrak{C}}([0,x]))^{\log_s p}}: x\in\mathfrak{C}\backslash\{0\}\right\}=[m,M]$ if $0\in A$, and $m=\frac{q(s-1)+r}{p-1}, M=\frac{q(p-1)+pr}{p-1}$ if the set $A$ consists of all the integers in $\{0,1,\ldots, p-1\}$ which have the same remainder $r\in\{0,1,\ldots,q-1\}$ modulus $q$ for some positive integer $q \geq 2$ (i.e. $h(x)=qx+r$). We further show that the sequence $\left\{\frac{a_n}{n^{\log_s p}}\right\}_{n\geq 1}$ is not uniformly distributed modulo 1, and it does not have the cumulative distribution function, but has the logarithmic distribution function (give by a specific Lebesgue integral).