Modular hyperbolas and Beatty sequences

TL;DR

This paper investigates bounds related to products in modular arithmetic constrained by Beatty sequences, using advanced exponential sum techniques and autocorrelation bounds, contributing new results under specific conditions on the sequence parameters.

Contribution

It provides new bounds for solutions to modular equations involving Beatty sequences, employing a method by Banks and Shparlinski and adapting Kloosterman sum techniques.

Findings

Bounds for $ ext{max}igrace{m, ilde{m}igrace}$ under given modular and sequence conditions

Average bounds for discrete autocorrelation of specific exponential sequences

Application of advanced exponential sum methods to Beatty sequence problems

Abstract

Bounds for $\max\{m,\tilde{m}\}$ subject to $m,\tilde{m} \in \mathbb{Z}\cap[1,p)$, $p$ prime, $z$ indivisible by $p$, $m\tilde{m}\equiv z\bmod p$ and $m$ belonging to some fixed Beatty sequence $\{ \lfloor n\alpha+\beta \rfloor : n\in\mathbb{N} \}$ are obtained, assuming certain conditions on $\alpha$. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence $0,\, \operatorname{e}_p(y\overline{1}), \operatorname{e}_p(y\overline{2}), \ldots, \operatorname{e}_p(y(\overline{p-1}))$ on average are obtained, where $\operatorname{e}_p(t) = \exp(2\pi i t/p)$ and $m\overline{m} \equiv 1\bmod p$. The latter is accomplished by adapting a method due to Kloosterman.