Notes and computations on forbidden differences

TL;DR

This paper investigates the maximum size of subsets of integers avoiding certain forbidden differences, providing new exact formulas, estimates, and computational data for specific cases like perfect squares and primes minus one.

Contribution

It offers new results including exact formulas and bounds for non-intersective difference sets, and presents the first computational data for maximum subset sizes up to N=500.

Findings

Exact formulas and estimates for D(X,N) in specific cases

Computational determination of D(S,N) for N≤300

Computational determination of D(ℙ−1,N) for N≤500

Abstract

We explore from several perspectives the following question: given $X\subseteq \mathbb{Z}$ and $N\in \mathbb{N}$, what is the maximum size $D(X,N)$ of $A\subseteq \{1,2,\dots,N\}$ before $A$ is forced to contain two distinct elements that differ by an element of $X$? The set of forbidden differences, $X$, is called \textit{intersective} if $D(X,N)=o(N)$, with the most well-studied examples being $X=S=\{n^2: n\in \mathbb{N}\}$ and $X=\mathcal{P}-1=\{p-1: p\text{ prime}\}$. In addition to some new results, including exact formulas and estimates for $D(X,N)$ in some non-intersective cases like $X=\mathcal{P}$ and $X=S+k$, $k\in \mathbb{N}$, we also provide a comprehensive survey of known bounds and extensive computational data. In particular, we utilize an existing algorithm for finding maximum cliques in graphs to determine $D(S,N)$ for $N\leq 300$ and $D(\mathcal{P}-1,N)$ for $N\leq 500$. None of these exact values appear previously in the literature.