Additive bases: change of domain

Journal article

We consider two questions of Ruzsa on how the minimum size of an additive basis B of a given set A depends on the domain of B. To state these questions, for an abelian group G and A⊆D⊆G we write ℓD(A):=min{|B|:B⊆D,A⊆B+B}. Ruzsa asked how much larger than ℓQ(A) can ℓZ(A) be for A⊂Z, and how much larger than ℓZ(A) can ℓN(A) be for A⊂N. For the first question we show that if ℓQ(A)=n then ℓZ(A)≤2n, and this is tight up to an additive error of at most O(n√). For the second question, we show that if ℓZ(A)=n then ℓN(A)≤O(nlogn), and this is tight up to the constant factor. We also consider these questions for higher order bases. Our proofs use some ideas that are unexpected in this context, including linear algebra and Diophantine approximation.

Authors: Bukh, B; van Hintum, P; Keevash, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematics
• Journal: Acta Arithmetica
• Publication date: 2025-10-28
• DOI: 10.4064/aa240912-24-9
• EISSN: 1730-6264
• ISSN: 0065-1036
• Language: English
• Keywords: Abelian group; Additive group; Constant (computer programming); Set (abstract data type); Diophantine equation; Domain (mathematical analysis); Mathematical proof