For any positive integer \(p\ge 3\) , let A be a proper subset of \(\{0,1,\ldots , p-1\}\) with \({\# }A=s\ge 2\) . Suppose \(h: \{0,1,\ldots ,s-1\}\rightarrow A\) is a strictly increasing one-to-one mapping. We focus on the so-called Cantor-integers \(\{a_n\}_{n\ge 1}\) , which consist of these positive integers n such that the digits of the p-ary expansion of n are taken solely from the elements of A. Evidently, the set \(\mathfrak {C}\) consisting of points \(x\in [0,1]\) such that the digits in the p-ary expansion of x are exclusively drawn from the elements of A is precisely the attractor of the iterated function system \(\{f_i\}_{i=0}^{s-1}\) , where \(f_i\) are functions defined on [0, 1] as \(f_i(x)=\frac{x+h(i)}{p}\) , respectively. Denote the classical self-similar measure supported on \(\mathfrak {C}\) by \(\mu _{\mathfrak {C}}\) with \(\mu _\mathfrak {C}=\sum _{i=0}^{s-1}\frac{1}{s} \mu _\mathfrak {C}\circ f_i^{-1}\) . Note that \(n^{\log _s p}\) is the growth order of \(a_n\) and the set of accumulation points of the sequence \(\left\{ \frac{a_n}{n^{\log _s p}}:~n\ge 1\right\} \) is precisely the set \(\left\{ \frac{x}{(\mu _{\mathfrak {C}}([0,x]))^{\log _s p}}: x\in \mathfrak {C}\cap [\frac{h(1)}{p},1]\right\} \) . We show that this set is just an interval with the infimum m and supremum M of the sequence \(\left\{ \frac{a_n}{n^{\log _s p}}:~n\ge 1\right\} \) as its endpoints, in particular, \(\left\{ \frac{x}{(\mu _{\mathfrak {C}}([0,x]))^{\log _s p}}: x\in \mathfrak {C}\backslash \{0\}\right\} =[m,M]\) if \(0\in A\) . Further, we show that the sequence \(\left\{ \frac{a_n}{n^{\log _s p}}\right\} _{n\ge 1}\) is not uniformly distributed modulo 1, and it does not have cumulative distribution function, but has logarithmic distribution function (given by a specific Lebesgue integral). In the special case where A is the set composed of numbers from \(\{0,1,\ldots , p-1\}\) that all have the same remainder of r when divided by some given positive integer \(q \ge 2\) (i.e. \(h(x)=qx+r\) ), we show that \(m=\frac{q(s-1)+r}{p-1}, M=\frac{q(p-1)+pr}{p-1}\) .
