Cardinalities of $g$-difference sets

TL;DR

This work resolves Kravitz’s question by proving that the limit $\lim_{n\to\infty}\frac{\eta_g(n)}{\sqrt{n}}$ exists and is positive for fixed $g$, and extends the analysis to vector spaces over finite fields where even- and odd-dimension limits $L_e$ and $L_o$ exist and are positive, with a conjectured distinction $L_e\neq L_o$. It also establishes the sharp asymptotic $\alpha_g(n)=(1+o_g(1))\sqrt{gn}$ for the maximal size of a set with at most $g$ representations of nonzero differences. The proofs blend refined Rédei–Rényi constructions, Singer’s theorem, Bose–Chowla Sidon-set methods, and number-theoretic input about primes in short intervals, yielding both integer and finite-field results and illuminating connections to coding theory and cryptography. Together, these results advance the understanding of additive bases and representation problems across groups and have implications for related combinatorial and algebraic structures.

Abstract

Let $η_{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 $\lim\limits_{n\rightarrow \infty}\frac{η_{g}(n)}{\sqrt{n}}$ exists, answering a question of Kravitz. We also investigate a similar problem in the setting of a vector space over a finite field. Let $α_g(n)$ 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 $α_g(n)={\sqrt{gn}}(1+o_{g}(1))$ as $n\rightarrow\infty$.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Conjecture 1
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof