arxiv: 2605.10771 · v1 · submitted 2026-05-11 · 🧮 math.CO · math.NT

Recognition: 2 theorem links

· Lean Theorem

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

Zixiang Xu

Pith reviewed 2026-05-12 04:23 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords additive basessumsetsreal vector spacesadditive combinatoricsgraph theorycoloring lemmasconjectures

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The pith

If S is any basis of R^n and S+S sits inside A+B with |A| at most n-t, then |B| must be at least n plus binom(t+1,2).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the strengthened size bound conjectured for additive bases over the rationals continues to hold when the ambient space is changed to the real numbers. For an arbitrary basis S of R^n, the containment S+S ⊆ A+B together with the upper bound |A| ≤ n-t forces |B| ≥ n + binom(t+1,2) for every admissible t, and the lower bound is attained by explicit constructions. A reader cares because the result gives an exact trade-off between the cardinalities of A and B while covering the doubled basis, and it settles the open extension question posed by Bukh. The short argument models the sumset condition inside a graph and applies a fresh coloring lemma over the two-element field.

Core claim

We prove the full strengthened statement over R^n: if S+S ⊆ A+B and |A| ≤ n-t with 0 ≤ t ≤ n-1, then |B| ≥ n + binom(t+1,2), which is sharp for every basis S and every 0 ≤ t ≤ n-1. The proof is short, using edge contractions in a graph-theoretical framework and a new coloring lemma over F_2^n.

What carries the argument

Graph modeling of the containment S+S ⊆ A+B, reduced via edge contractions to a new coloring lemma on F_2^n.

If this is right

The bound is attained for explicit choices of A and B for every basis S.
The original conjecture over Q^n follows immediately as a special case.
When t=0 the result forces |A|+|B| ≥ 2n whenever A+B covers S+S.
The same minimal sizes are forced for every choice of basis S.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction to an F_2^n coloring problem may adapt to sumset questions in other abelian groups.
Similar strengthened bounds could be sought for higher-order sumsets such as S+S+S.
The sharpness constructions may supply minimal covering sets in other vector-space problems.

Load-bearing premise

S must be a linearly independent spanning set for R^n, since this property is required for the sumset condition to translate into a graph on which the F_2^n coloring lemma applies.

What would settle it

A concrete counterexample would be any basis S of R^n, any set A of size n-t, and any set B with fewer than n + binom(t+1,2) elements such that S+S is still contained in A+B.

read the original 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 \(\mathbb F_2^n\).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Xu proves the strengthened Bukh-van Hintum-Keevash conjecture over the reals for arbitrary bases with a short graph-contraction argument and new F_2^n coloring lemma.

read the letter

The key takeaway is that this paper proves the strengthened conjecture over R^n for any basis S: when S+S sits inside A+B and |A| is at most n-t, |B| must be at least n plus binom(t+1,2), with matching constructions for sharpness. Xu brings in a graph model where basis elements are vertices, does edge contractions, and uses a fresh coloring lemma on F_2^n to control the sizes. This moves past the rational setting in the original work and gives a direct proof without fitting parameters. The shortness of the argument is a real strength here, as it keeps the logic transparent. The techniques seem well-suited to the problem, and the claim of sharpness for every basis adds credibility. The derivation avoids self-reference and builds straight from the assumptions. A minor point is that the coloring lemma's details are crucial, and while the abstract suggests it works cleanly, one would want to see exactly how the F_2^n coloring translates back to the real space without losing the linear independence. But the stress-test found no inconsistencies, so this is probably fine. Readers in additive combinatorics, especially those interested in basis sumsets or extremal problems in vector spaces, will get the most from this. It is a targeted advance rather than a broad overhaul. I would send this to peer review. The resolution of the conjecture with new tools makes it worth the referees' time, even if some polishing on the lemma presentation could help.

Referee Report

0 major / 1 minor

Summary. The manuscript proves the strengthened form of the Bukh-van Hintum-Keevash conjecture over the reals: for any basis S of R^n and subsets A,B of R^n satisfying S+S ⊆ A+B with |A| ≤ n-t (0 ≤ t ≤ n-1), one has |B| ≥ n + binom(t+1,2). The bound is shown to be sharp by explicit constructions that work for every basis S. The argument models the sumset condition as a graph on the basis vectors, performs edge contractions, and invokes a new coloring lemma over F_2^n.

Significance. If the proof is correct, the result fully resolves the strengthened conjecture in the general real-vector-space setting with arbitrary bases, extending the original rational case. The short, direct derivation (no free parameters, no circular reductions) and the matching sharpness examples for all admissible t constitute a clean contribution to additive combinatorics. The new F_2^n coloring lemma is a reusable tool whose introduction strengthens the paper.

minor comments (1)

A one-sentence informal statement of the new coloring lemma in the abstract or introduction would improve readability for readers who do not immediately consult the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending acceptance. We appreciate the recognition of the result's significance, the shortness of the proof, and the utility of the new coloring lemma.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper supplies an explicit short proof of the strengthened bound for arbitrary bases S of R^n. It models the sumset condition S+S ⊆ A+B as a graph on basis elements, performs edge contractions, and invokes a new coloring lemma over F_2^n that is stated and proved within the manuscript. No load-bearing step reduces by definition to the target bound, no parameter is fitted and renamed as a prediction, and no self-citation chain is used to justify the central claim. The sharpness constructions are explicit and independent of the proof. This matches the default expectation for a self-contained combinatorial proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proof rests on standard vector-space axioms and graph theory; the only novel element is the coloring lemma proved inside the paper.

axioms (2)

standard math R^n forms a vector space over the reals under componentwise addition and scalar multiplication
Required to define bases S and the sumset A+B
domain assumption Standard notions of graphs, edge contraction, and vertex coloring apply to the auxiliary graph constructed from the sumset condition
Central to the proof framework

invented entities (1)

New coloring lemma over F_2^n no independent evidence
purpose: To bound the size of B after graph contractions
The lemma is introduced and proved within the paper to complete the argument

pith-pipeline@v0.9.0 · 5532 in / 1422 out tokens · 70220 ms · 2026-05-12T04:23:17.357651+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We construct a bipartite graph G ... contract all diagonal edges εii ... color the vertices of H by elements of the quotient group R^n / 2Z^n ... Lemma 2.2 ... Dn ⊆ X−C
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
if S+S ⊆ A+B and |A| ≤ n−t ... |B| ≥ n + binom(t+1,2)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

N. Alon, B. Bukh, and B. Sudakov. Discrete Kakeya-type problems and small bases.Israel J. Math., 174:285–301, 2009

work page 2009
[2]

B. Bukh. Problems.https://www.borisbukh.org/problems.html

work page
[3]

B. Bukh, P. van Hintum, and P. Keevash. Additive bases: change of domain.Acta Arith., 221(3):239–252, 2025

work page 2025
[4]

J.-R. Chen. Waring’s problem forgp5q “37.Sci. Sinica, 13:335, 1964

work page 1964
[5]

J.-R. Chen. On the representation of a larger even integer as the sum of a prime and the product of at most two primes.Sci. Sinica, 16:157–176, 1973

work page 1973
[6]

Erd˝ os and D

P. Erd˝ os and D. J. Newman. Bases for sets of integers.J. Number Theory, 9(4):420–425, 1977

work page 1977
[7]

H. A. Helfgott. The ternary Goldbach conjecture is true.Ann. of Math., 182(1):1–72, 2015

work page 2015
[8]

D. Hilbert. Beweis f¨ ur die darstellbarkeit der ganzen zahlen durch eine feste anzahln-ter potenzen. Math. Ann., 67(3):281–300, 1909

work page 1909
[9]

M. B. Nathanson.Additive number theory: The classical bases, volume 164 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1996

work page 1996
[10]

I. Ruzsa. Open problem session. Additive Combinatorics and Fourier Analysis Workshop, Budapest, June 2024

work page 2024
[11]

Tao and V

T. Tao and V. Vu.Additive combinatorics, volume 105 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006

work page 2006
[12]

I. M. Vinogradov. Representation of an odd number as a sum of three primes.Doklady Akademii Nauk SSSR, 15:291–294, 1937. 5

work page 1937