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

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).

• The paper proves that for any basis S of ℝⁿ, S+S⊆A+B and |A|≤n−t imply |B|≥n+binom(t+1,2), resolving the strengthened conjecture sharply.
• The proof reduces to the standard basis, encodes sum representations in a contracted bipartite graph, and applies a new coloring lemma over 𝔽₂ⁿ.
• The result holds for every basis and all 0≤t≤n−1, and confirms that the factor 2 in the rational-to-integer change-of-domain theorem for 2-bases is optimal.

Background and motivation

A central theme in additive combinatorics is the efficiency with which a set can be represented by sumsets. In the finite setting initiated by Erdős and Newman, one asks how small a set $B$ can be while covering a target set $A$ via sums, and this line of inquiry has connections to discrete Kakeya-type phenomena and to Ruzsa's questions on change-of-domain for additive bases. Bukh, van Hintum and Keevash studied the change-of-domain problem for $2$-bases: how much larger must an integer $2$-basis be compared with a rational one? They proved that every rational $2$-basis of size $n$ can be replaced by an integer one of size at most $2n$, with the factor $2$ tight up to lower-order error. A key ingredient of their reduction was a vector model: minimize $|A|+|B|$ subject to $A,B\subseteq\mathbb{Q}^n$ and $A$0, for which they established the lower bound $A$1 and conjectured the sharp value $A$2, together with the refinement that $A$3 should force $A$4. Bukh further asked whether the same bounds hold over $A$5 for an arbitrary basis $A$6 with $A$7.

Main result

The paper proves the full strengthened statement over the reals.

Theorem. Let $A$8 be a basis of $A$9 and $2$0 with $2$1. If $2$2 for some $2$3, then $2$4.

This resolves both the original conjecture of Bukh, van Hintum and Keevash (the case $2$5 gives $2$6) and its refined strengthening, and does so for arbitrary bases of $2$7 rather than only the standard basis of $2$8. The bound is sharp for every basis $2$9 and every $2$0: taking $2$1 and

$2$2

gives $2$3 with $2$4; distinctness of the displayed elements follows from linear independence. A direct corollary is the conjectured bound over $2$5 for the standard basis, and combined with the reduction in prior work this shows the factor $2$6 in the change-of-domain theorem is best possible already for $2$7-bases: for every $2$8 there is a set of integers with a rational $2$9-basis of size at most $2$0 whose smallest integer $2$1-basis has size exactly $2$2.

Proof architecture

The proof proceeds in three stages: a reduction to the standard basis, a graph-theoretic encoding, and a counting argument via a new coloring lemma.

Reduction. Since any basis $2$3 is the image of the standard basis under an invertible linear map, and such maps preserve cardinalities and sumset inclusions, it suffices to handle $2$4 (the case $2$5 being trivial).

Graph construction. Fix one representation $2$6 for each pair $2$7, and build a bipartite multigraph $2$8 on copies of the used sets $2$9 (size $n$0) and $n$1 (size $n$2), with one edge $n$3 per pair. The map $n$4 sending left vertices to their vector and right vertices to the negation satisfies $n$5. The diagonal edges $n$6 form a forest: a cycle would yield a nontrivial signed linear relation $n$7 among distinct basis vectors, contradicting linear independence. Contracting the $n$8 diagonal edges therefore reduces the vertex count by exactly $n$9, giving $2n$0 for the contracted graph $2n$1.

Coloring. Vertices of $2n$2 are colored by elements of $2n$3 via $2n$4; this is well-defined because vertices merged by diagonal-edge contraction have $2n$5-values differing by elements of $2n$6. Let $2n$7 be the colors appearing on vertices containing a left vertex; then $2n$8, and each off-diagonal edge forces $2n$9, where $2$0 is the full color set.

The coloring lemma

The combinatorial core is the following lemma over $2$1.

Lemma. Let $2$2 be a subgroup of an abelian group $2$3, $2$4. If $2$5 with $2$6 and $2$7, then $2$8.

The proof first treats the case $2$9. After translating so that $|A|+|B|$0, with $|A|+|B|$1 and $|A|+|B|$2 of dimension at most $|A|+|B|$3, the quotient map $|A|+|B|$4 satisfies $|A|+|B|$5 (since $|A|+|B|$6 collapses to the zero coset and the quotient has exponent $|A|+|B|$7). Choosing $|A|+|B|$8 indices whose $|A|+|B|$9-images form a basis of the quotient, the $A,B\subseteq\mathbb{Q}^n$0 pairwise sums of these basis vectors are distinct nonzero elements of $A,B\subseteq\mathbb{Q}^n$1, so $A,B\subseteq\mathbb{Q}^n$2.

The reduction from general $A,B\subseteq\mathbb{Q}^n$3 uses a folding argument: since every element of $A,B\subseteq\mathbb{Q}^n$4 lies in $A,B\subseteq\mathbb{Q}^n$5, any representation $A,B\subseteq\mathbb{Q}^n$6 uses elements from a common $A,B\subseteq\mathbb{Q}^n$7-coset, so translating each coset meeting $A,B\subseteq\mathbb{Q}^n$8 back into $A,B\subseteq\mathbb{Q}^n$9 preserves $A$00, does not increase either cardinality, and preserves the containment $A$01.

Applying the lemma with $A$02, $A$03, $A$04, $A$05, $A$06 yields $A$07. Chaining $A$08 gives $A$09, and since $A$10,

$A$11

completing the proof. The identity $A$12 is precisely where the refined quadratic term arises from the coloring count.

Remarks on scope and limitations

The result is exact and sharp across the full range $A$13, for every basis of $A$14, with no hidden assumptions beyond finiteness of $A$15 (the infinite case being trivial). The method is specific to $A$16-bases: the argument exploits the exponent-$A$17 structure of $A$18 and the pairwise-sum set $A$19, and does not directly address $A$20-bases for $A$21, nor the original change-of-domain question beyond the $A$22-basis setting. The paper also does not determine whether the vector-model bound extends to other ground fields or quotient structures; these remain open questions raised by the framework rather than settled by it.

Conclusion

The paper resolves the strengthened conjecture of Bukh, van Hintum and Keevash in full generality over $A$23, proving that $A$24 with $A$25 forces $A$26, with matching constructions for every basis and every $A$27. The proof combines a forest-contraction argument on a bipartite multigraph with a coloring lemma over $A$28, and as a corollary establishes the sharpness of the factor $A$29 in the rational-to-integer change-of-domain theorem for $A$30-bases.