Linear Diophantine equations in Piatetski-Shapiro sequences

TL;DR

This paper investigates the solutions of linear Diophantine equations within Piatetski-Shapiro sequences, establishing conditions for infinite solutions and analyzing the Hausdorff dimension of relevant parameter sets.

Contribution

It characterizes the set of exponents  for which the equation has infinitely many solutions in Piatetski-Shapiro sequences and provides bounds on the Hausdorff dimension of this set.

Findings

Identifies  with infinitely many solutions for the equation.

Provides a lower bound for the Hausdorff dimension of the set of such .

Shows existence of uncountably many  > 2 with infinitely many 3-term arithmetic progressions.

Abstract

A Piatetski-Shapiro sequence with exponent $\alpha$ is a sequence of integer parts of $n^\alpha$ $(n = 1,2,\ldots)$ with a non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set of those terms. In this article, we study the set of $\alpha$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $\alpha > 2$ such that $\mathrm{PS}(\alpha)$ contains infinitely many arithmetic progressions of length $3$.