bibkey: nathanson2026hbases
authors: Melvyn B. Nathanson
year: 2026
title: "Problems in additive number theory, VII: The structure of additive h-bases for n"
doi: null
url: https://arxiv.org/abs/2605.26425v3
claim: "Problem 12 asks whether negative elements force ell_h(A) <= n-flat_h(k), or even the strict inequality ell_h(A) < n-flat_h(k)."
strata_touched:
  - D5/S3/Arith/NathansonAdditiveHBasisRefutation
license: citation-only
triage: anchor

Negative elements in finite additive segment bases

For an integer set A, the paper defines hA as the set of sums of exactly h not necessarily distinct elements of A. It calls A an additive h-basis for n when [0,n] is contained in hA, and defines ell_h(A) as the largest such n. The quantity is undefined when 0 is not in hA.

For positive h and k, the paper defines n-flat_h(k) as the largest covered endpoint among all k-element sets of nonnegative integers. Problem 12 asks, for h >= 2, k >= 2, and a k-element integer set with negative minimum, whether ell_h(A) <= n-flat_h(k) and whether the strict inequality ell_h(A) < n-flat_h(k) holds. Only the strict statement is addressed by the associated result.

The reflection example on printed page 7 gives B={-1,1,2} and 2B={-2,0,1,2,3,4}. The source uses this set to illustrate reflection of interval bases, not as a stated answer to Problem 12.

Verified locator

• Source: https://arxiv.org/abs/2605.26425v3
• Definitions of hA, additive h-basis, and ell_h(A): printed page 2.
• Definition of n-flat_h(k): printed pages 7-8.
• Problem 12: printed page 9.