通用$β$ -扩展
Universal $β$-expansions
摘要：
给定$β\in(1,2)$，实数$x$的$β$展开式是以$β$为基的幂级数，系数为0和1，其和等于$x$。本文的目的是研究与$β$ -展开式的普适性和组合学有关的某些问题。我们的主要结果是，对于任何$β\in(1,2)$和a.e. $x\in (0,1)$总是存在一个通用的$β$ -在Erdös和Komornik意义上的$x$的扩展，即，$β$ -扩展，其复杂度函数是$2^n$。我们还研究了一些问题，有关的点有小于一个完整的分支连续的$β$ -扩展，也正常的$β$ -扩展。
英文摘要：
Given $β\in(1,2)$, a $β$-expansion of a real $x$ is a power series in base $β$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of $β$-expansions. Our main result is that for any $β\in(1,2)$ and a.e. $x\in (0,1)$ there always exists a universal $β$-expansion of $x$ in the sense of Erdös and Komornik, i.e., a $β$-expansion whose complexity function is $2^n$. We also study some questions related to the points having less than a full branching continuum of $β$-expansions and also normal $β$-expansions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
