Distribution of squares modulo a composite number

TL;DR

This paper investigates how squares are distributed modulo square-free numbers, explores inverse problems related to the large sieve, and improves existing results on the distribution of residue tuples.

Contribution

It provides new insights into the distribution patterns of squares modulo composite numbers and enhances bounds for the distribution of s-tuples of reduced residues.

Findings

Distribution of squares modulo square-free numbers characterized

Improved bounds on the distribution of s-tuples of reduced residues

Enhanced understanding of inverse large sieve problems

Abstract

In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of $s$-tuples of reduced residues.