Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences

2023/06/30 by Saito, Kota
#11D04 #11K55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.17813

Abstract

A sequence of integers of the form \lfloor n^α\rfloor (n=1,2,…) for some fixed non-integral α>1 is called a Piatetski-Shapiro sequence, where \lfloor x\rfloor denotes the integer part of x. Let PS(α) denote the set of all those terms. In this article, we show that x+y=z has only finitely many solutions (x,y,z)∈ PS(α)³ for almost every α>3. Furthermore, we show that PS(α) has only finitely many arithmetic progressions of length 3 for almost every α>10. In addition, we estimate upper bounds for the Hausdorff dimension of the set of α∈ [s,t] such that y=a₁x₁+⋯ +a_nx_n has infinitely many solutions on PS(α).

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