A solution to a strengthened conjecture of Bukh, van Hintum and Keevash on additive bases
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909033274802176 |
|---|---|
| author | Xu, Zixiang |
| author_facet | Xu, Zixiang |
| contents | 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 \(\mathbb F_2^n\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10771 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A solution to a strengthened conjecture of Bukh, van Hintum and Keevash on additive bases Xu, Zixiang Combinatorics Number Theory 11B13 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 \(\mathbb F_2^n\). |
| title | A solution to a strengthened conjecture of Bukh, van Hintum and Keevash on additive bases |
| topic | Combinatorics Number Theory 11B13 |
| url | https://arxiv.org/abs/2605.10771 |