论文标题
关于插座者的分布
On the distribution of the Cantor-integers
论文作者
论文摘要
对于任何正整数$ p \ geq 3 $，让$ a $为$ \ {0,1，\ ldots，p-1 \} $的适当子集，带有$ \ sharp a = s \ geq 2 $。假设$ h：\ {0,1，\ ldots，s-1 \} \ to a $是一对一地图，严格增加了$ a = \ {h（h（0），h（1），\ ldots，h（s-1）$。我们专注于所谓的cantor-Integers $ \ {a_n \} _ {n \ geq 1} $，其中由这些正整数$ n $组成，因此$ p $ n $的所有数字$ n $均属于$ a $。令$ \ mathfrak {c} = \ left \ {\ sum \ limits_ {n \ geq 1} \ frac {\ varepsilon_n} {p^n} {p^n}：\ varepsilon_n_n \ in a \ text {for a protitial integer} n \ right as right persim and complor persim and complor persim and comport \ right as inim and coptor contor contor contor contor contor， $ \ mathfrak {c} $ by $μ_ {\ mathfrak {c}} $。现在，$ n^{\ log_s p} $是$ a_n $和$ \ left \ weft \ {\ frac {a_n} {n^{n^{\ log_s p}}的增长顺序： $ \ left \ {\ frac {x} {（μ_ {\ mathfrak {c}}}}（[0，x]）） $ e $，我们表明$ \ weft \ {\ frac {a_n} {n^{\ log_s p}}}}：〜n \ geq 1 \ right \}'$只是一个间隔$ [m，m] $，带有$ m：= = \ inf \ eft \ eft \ left \ left \ eft \ frac {\ frac { p}}：n \ geq 1 \ right \} $ and $ m：= \ sup \ left \ {\ frac {a_n} {n^{n^{\ log_s p}}}：n \ geq 1 \ right \} $。特别是，$ \ left \ {\ frac {x} {（μ_ {\ mathfrak {c}}}（c}}（[0，x]）） $ m = \ frac {q（s-1）+r} {p-1}，m = \ frac {q（p-1）+pr} {p-1} $如果集合$ a $由$ \ {0,1，\ ldots，p-1 \} $中的所有整数组成，这些整数在$ \ {0,1，\ ldots，p-1 \} $中，这些整数具有相同的保留，这些整数具有相同的$ r \ rd \ rd \ rd \ c in n int int int int int int某些正整数$ q \ geq 2 $的模量$ q $（即$ h（x）= qx+r $）。我们进一步表明，序列$ \ weft \ {\ frac {a_n} {n^{\ log_s p}}} \ right \} _ {n \ geq 1} $不是均匀分布的modulo 1，并且没有累积的分布功能，但具有累积的分布函数，而不是一个特定的分布函数（由一个特定的分布函数（missecte ande conbere）。
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 $μ_{\mathfrak{C}}$. 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 $\left\{\frac{x}{(μ_{\mathfrak{C}}([0,x]))^{\log_s p}}: x\in\mathfrak{C}\cap[\frac{h(1)}{p},1]\right\}$, 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, $\left\{\frac{x}{(μ_{\mathfrak{C}}([0,x]))^{\log_s p}}: x\in\mathfrak{C}\backslash\{0\}\right\}=[m,M]$ 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).
