The maximal density of product-free sets in Z/nZ

TL;DR

This paper investigates the maximum possible density of product-free sets in modular arithmetic and establishes bounds that closely match, advancing understanding of their size in large residue classes.

Contribution

It provides tight asymptotic bounds on the maximal density of product-free sets in Z/nZ as n grows large, improving previous results.

Findings

Established upper bounds on the density of product-free sets

Constructed sets with densities approaching the upper bounds

Matched bounds up to constants for large n

Abstract

This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab == c (mod n) with a,b,c in the set. In a previous paper we constructed an infinite sequence of integers (n_i)_{i > 0} and product-free sets S_i in Z/n_iZ such that the density |S_i|/n_i tends to 1 as i tends to infinity, where |S_i|$ denotes the cardinality of S_i. Here we obtain matching, up to constants, upper and lower bounds on the maximal attainable density as n tends to infinity.