A solution to a strengthened conjecture of Bukh, van Hintum and Keevash on additive bases

TL;DR

This paper proves a strengthened conjecture on additive bases in real vector spaces, establishing sharp bounds on the size of one set given the other, using graph theory and a new coloring lemma.

Contribution

It provides the full proof of a strengthened conjecture for additive bases in \\mathbb{R}^n, extending previous results and confirming sharp bounds for all bases.

Findings

Proved the strengthened conjecture for additive bases in \\mathbb{R}^n.

Established sharp bounds on set sizes based on the basis and subset sizes.

Introduced a novel coloring lemma over \\mathbb{F}_2^n and used graph-theoretical methods.

Abstract

Motivated by the change-of-domain problem for additive bases, Bukh, van Hintum and Keevash conjectured that if \(A,B\subseteq \mathbb{Q}^{n}\) and \(\{\boldsymbol{e}_i+\boldsymbol{e}_j:1\le i\le j\le n\}\subseteq A+B,\) then \(|A|+|B|\ge 2n\). They further proposed the strengthened conjecture: if \(|A|=n-t\), then \(|B|\ge n+\binom{t+1}{2}.\) Bukh also explicitly asked whether the same bounds hold for \(A,B\subseteq \mathbb{R}^{n}\) and an arbitrary basis \(S\) of \(\mathbb{R}^{n}\), under the assumption \(S+S\subseteq A+B\). We prove the full strengthened statement over \(\mathbb{R}^{n}\): if \(S+S\subseteq A+B\) and \(|A|\le n-t\) with \(0\le t\le n-1\), then \(|B|\ge n+\binom{t+1}{2},\) which is sharp for every basis \(S\) and every \(0\le t\le n-1.\) The proof is short, using edge contractions in a graph-theoretical framework and a new coloring lemma over \(\mathbb F_2^n\).