具有简单正规$β$ -展开式的数
Numbers with simply normal $β$-expansions
摘要：
在[巴克]的第一作者证明，任何$β\in (1,β_{KL})$每$x\in(0,\frac{1}{β-1})$有一个简单的正常$β$ -扩展，其中$β_{KL}\approx 1.78723$是Komornik-Loreti常数。[JSS]中的观察结果补充了这一结果，其中表明，无论何时$β\in (β_T, 2]$，都存在具有唯一$β$展开的$x\in(0,\frac{1}{β-1})$，并且这种展开不是简单的正态展开。这里，$β_T\approx 1.80194$是多项式$x^3-x^2-2x+1$的$(1,2]$中的唯一零点。这在我们对间隔$[β_{KL}, β_T]$的理解中留下了空白。在本文中，我们填补了这一空白，并证明了任何$β\in (1,β_T],$的每一个$x\in(0,\frac{1}{β-1})$有一个简单的正常$β$ -扩展。为了完成，我们提供了一个证明，任何$β\in(1,2)$，勒贝格几乎每一个$x$有一个简单的正常$β$ -扩展。我们还给出了$x$与多个$β$ -扩展的例子，其中没有一个是简单的正常。我们的证明依赖于组合数学中关于单词和动力系统的思想。
英文摘要：
In [Bak] the first author proved that for any $β\in (1,β_{KL})$ every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion, where $β_{KL}\approx 1.78723$ is the Komornik-Loreti constant. This result is complemented by an observation made in [JSS], where it was shown that whenever $β\in (β_T, 2]$ there exists an $x\in(0,\frac{1}{β-1})$ with a unique $β$-expansion, and this expansion is not simply normal. Here $β_T\approx 1.80194$ is the unique zero in $(1,2]$ of the polynomial $x^3-x^2-2x+1$. This leaves a gap in our understanding within the interval $[β_{KL}, β_T]$. In this paper we fill this gap and prove that for any $β\in (1,β_T],$ every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion. For completion, we provide a proof that for any $β\in(1,2)$, Lebesgue almost every $x$ has a simply normal $β$-expansion. We also give examples of $x$ with multiple $β$-expansions, none of which are simply normal. Our proofs rely on ideas from combinatorics on words and dynamical systems.
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。
