Arithmetical properties of real numbers related to beta-expansions

TL;DR

This paper investigates the arithmetical properties of certain infinite sums involving Pisot or Salem numbers and develops criteria for algebraic and linear independence of these sums, with applications to specific sequences.

Contribution

It introduces new criteria for algebraic and linear independence of sums related to beta-expansions, applicable to sequences with specific growth conditions.

Findings

Proved algebraic independence of sums involving m^{log m} and m^{log log m}

Established criteria for linear independence for sums with exponents m^ ho, ho > 1

Applied criteria to sums involving beta and specific sequences

Abstract

The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} \beta^{-w(m)}\), where \(\beta\) 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 \[\sum_{m=1}^{\infty}\beta^{-\lfloor \varphi(1,0;m)\rfloor}, \sum_{m=3}^{\infty}\beta^{-\lfloor \varphi(0,1;m)\rfloor}\] are algebraically independent, where \(\varphi(1,0;m)=m^{\log m}\) and \(\varphi(0,1;m)=m^{\log\log m}\). \par Moreover, we also give criteria for linear independence of real numbers. Our criteria are applicable to the values \(\sum_{m=0}^{\infty}\beta^{-\lfloor m^\rho\rfloor}\), where \(\beta\) is a Pisot or Salem number and \(\rho\) is a real number greater than 1.