论文标题
限制一类Cantor-Integers的行为
Limit behavior of a class of Cantor-integers
论文作者
论文摘要
在本文中，我们研究了一类cantor-integers $ \ {c_n \} _ {n \ geq 1} $，其中基本转换函数$ f：\ {0，\ dots，m \} \ to \ to \ to \ {0，\ dots，\ dots，d dots，p \} $严格增加和满足$ f（$ f（$ f（$）$ f（$ f（$ f（0）$ f（$ f（$ f（0）= 0）首先，我们提供算法来计算顺序$ \ weft \ {\ frac {c_n} {n^α} \ right \} _ { 功能。其次，我们证明了序列$ \ weft \ {\ frac {c_n} {n^α} \ right \} _ {n \ geq 1} $在紧密的间隔中密集，终点分别为其下点和优点。结果，（i）我们在$ \ mathfrak {c} $ $ 0 $上支持的自相似度量的上点和下点密度$ 1/α$密度，其中$ \ mathfrak {c} $是由Cantor-Integers引起的cantor集。 （ii）序列$ \ {\ frac {c_n} {n^α} \} _ {n \ geq 1} $没有累积的分布函数，但具有对数分布函数（由特定的lebesgue积分给出）。最后，我们获得了用于插管者的总和函数的Mellin-Perron公式。此外，我们研究了Cantor-Integer引起的极限函数的一些分析性能。
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 $\left\{\frac{C_n}{n^α}\right\}_{n\geq 1}$ where $α=\log_{m+1}^{p+1}$, and obtain the exact values of the superior and inferior when $f$ is a class of quadratic function. Secondly we show that the sequence $\left\{\frac{C_n}{n^α}\right\}_{n\geq 1}$ 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 $1/α$-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 $\{\frac{C_n}{n^α}\}_{n\geq 1}$ 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.
