Limit behavior of a class of Cantor-integers

Abstract: In this paper, we study a class of Cantor-integers ${C_n}{n\geq 1}$ with the base conversion function $f:{0,\dots,m}\to {0,\dots,p}$ being strictly increasing and satisfying $f(0)=0$ and $f(m)=p$. Firstly we provide an algorithm to compute the superior and inferior of the sequence $\left{\frac{C_n}{n^{\alpha}}\right}{n\geq 1}$ where $\alpha =\log_{m+1}^{p+1}$, and obtain the exact values of the superior and inferior when $f$ is a class of quadratic function. Secondly we show that the sequence $\left{\frac{C_n}{n^{\alpha}}\right}_{n\geq 1}$ is dense in the close interval with the endpoints being its inferior and superior respectively. As a consequence, (i) we get the upper and lower pointwise density $1/\alpha$-density of the self-similar measure supported on $\mathfrak{C}$ at $0$, where $\mathfrak{C}$ is the Cantor set induced by Cantor-integers. (ii) the sequence ${\frac{C_n}{n^{\alpha}}}_{n\geq 1}$ does not have cumulative distribution function but have logarithmic distribution functions (given by a specific Lebesgue integral). Lastly we obtain the Mellin-Perron formula for the summation function of Cantor-integers. In addition, we investigate some analytic properties of the limit function induced by Cantor-integers.