The largest $(k, \ell)$-sum-free subsets

TL;DR

This paper investigates the size of the largest $(k,\, ext{ extltilde})$-sum-free subsets within sets of positive integers, determining key constants and confirming conjectures for infinitely many cases in additive combinatorics.

Contribution

The paper determines the constant $c(k,\ell)$ for all $(k,\ell)$-sum-free sets and verifies the conjecture for infinitely many such sets, advancing understanding in additive combinatorics.

Findings

Determined the constant $c(k,\ell)$ for all $(k,\ell)$-sum-free sets.

Confirmed the conjecture for infinitely many $(k,\ell)$ cases.

Extended the analysis to restricted $(k,\ell)$-sum-free sets.

Abstract

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell)$-sum-free sets and restricted $(k,\ell)$-sum-free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$-sum-free sets, and confirm the conjecture for infinitely many $(k,\ell)$.