Cardinalities of $g$-difference sets

TL;DR

This paper investigates the minimal and maximal sizes of sets in the integers and finite fields that satisfy specific difference properties, establishing the existence of certain limits and asymptotic formulas.

Contribution

It proves the existence of the limit of the normalized minimal size of g-difference bases and derives asymptotic behavior for the maximal size of sets with bounded difference solutions.

Findings

The limit of η_g(n)/√n exists as n→∞.

α_g(n) asymptotically equals √(gn) as n→∞.

Provides new bounds and asymptotic formulas for difference set cardinalities.

Abstract

Let $\eta_{g}(n) $ be the smallest cardinality that $A\subseteq {\mathbb Z}$ can have if $A$ is a $g$-difference basis for $[n]$ (i.e, if, for each $x\in [n]$, there are {\em at least} $g$ solutions to $a_{1}-a_{2}=x$ ). We prove that the finite, non-zero limit $\lim\limits_{n\rightarrow \infty}\frac{\eta_{g}(n)}{\sqrt{n}}$ exists, answering a question of Kravitz. We also investigate a similar problem in the setting of a vector space over a finite field. Let $\alpha_g(n)$ be the largest cardinality that $A\subseteq [n]$ can have if, for all nonzero $x$, $a_{1}-a_{2}=x$ has {\em at most} $g$ solutions. We also prove that $\alpha_g(n)={\sqrt{gn}}(1+o_{g}(1))$ as $n\rightarrow\infty$.