一类Cantor整数的渐近性质

Asymptotic Behaviour of a Class of Cantor-integers

对于任意的正整数p≥3,用{a_(n)}_(n≥0)表示p-进展式数字只取偶数的非负整数所构成的数列.我们给出了a_(n)的增长阶为log_(s)p,其中s=[p/2],[·]为取上整函数.证明了{an/nlogsp}_(n≥1)在[2s-2/p-1,2]中稠密.并从测度的角度对该稠密性加以阐释.

For any integer p≥3,let {a}denotes the sequence of non-negative integers whose digits in the p-ary expansion are even.We show that the growth order of ais logp,where s=[p/2],[·] is the ceiling function.Moreover, {a/n}is dense in [2 s-2/p-1,2].As a consequence,an interesting result about the self-similar measure is established.