arXiv 1708.030932017-08-10math.NT

与β-展开式有关的实数的算术性质

Arithmetical properties of real numbers related to beta-expansions

Hajime Kaneko

备注：: 34 pages

摘要：

本文的主要目的是研究值\(\sum_{m=0}^{\infty} β^{-w(m)}\)的算术性质，其中\(β\)是一个固定的Pisot数或Salem数，\(w(m)\)（\(m=0,1,\ldots\)）是不同的非负整数序列，对任意大的\(m\)，有\(w(m+1)>w(m)\).我们首先介绍的标准，这些值的代数独立性。我们的准则适用于某些序列\(w(m)\)（\(m=0,1,\ldots\)）与\(\lim_{m\to\infty}w(m+1)/w(m)=1.\)例如，我们证明了两个数字\[\sum_{m=1}^{\infty}β^{-\lfloor φ(1,0;m)\rfloor}, \sum_{m=3}^{\infty}β^{-\lfloor φ(0,1;m)\rfloor}\]是代数独立的，其中\(φ(1,0;m)=m^{\log m}\)和\(φ(0,1;m)=m^{\log\log m}\)。此外，我们还给出了实数线性无关的判别准则。我们的准则适用于值\(\sum_{m=0}^{\infty}β^{-\lfloor m^ρ\rfloor}\)，其中\(β\)是Pisot或Salem数，\(ρ\)是大于1的实数。

英文摘要：

The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} β^{-w(m)}\), where \(β\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce criteria for the algebraic independence of such values. Our criteria are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}β^{-\lfloor φ(1,0;m)\rfloor}, \sum_{m=3}^{\infty}β^{-\lfloor φ(0,1;m)\rfloor}\] are algebraically independent, where \(φ(1,0;m)=m^{\log m}\) and \(φ(0,1;m)=m^{\log\log m}\). \par Moreover, we also give criteria for linear independence of real numbers. Our criteria are applicable to the values \(\sum_{m=0}^{\infty}β^{-\lfloor m^ρ\rfloor}\), where \(β\) is a Pisot or Salem number and \(ρ\) is a real number greater than 1.

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。

摘要：

英文摘要：

相关论文