bibkey: bhanja2025distinctterms
authors: Jagannath Bhanja
year: 2025
title: Sumsets with a minimum number of distinct terms
doi: null
url: https://arxiv.org/abs/2307.03977v2
claim: Question 1 asks whether ordinary and at-most-r-distinct h-fold sumsets have equal minimum cardinalities over k-element subsets of every abelian group.
strata_touched:
  - D5/S3/FiniteGroups/BhanjaRestrictedSumset/Result
license: citation-only
triage: anchor

Sumsets with a minimum number of distinct terms

J. Bhanja, Integers 25 (2025), A12, arXiv:2307.03977v2. The at-most-distinct restriction permits repetition and limits the number of different summand values, rather than requiring all summands distinct.

Verified locator

Journal: https://math.colgate.edu/~integers/z12/z12.pdf

Source: https://arxiv.org/abs/2307.03977v2

§4, “Further Remarks”, printed p. 11. Verbatim text fragments from the arXiv v2 text extraction (the superscript is extracted as (≤r)):

Another sumset that may come immediately to the reader’s mind is the sumset h(≤r) A, which contains those elements of the group G that are the sum of h elements of A with at most r distinct elements. The sumset hA is a special case of h(≤r) A for r = h.

Question 1. Is it always the case that the minimum size of both the sumsets hA and h(≤r) A are the same in any abelian group G, where the minimum runs over the subsets of G of size (say) k?

The preceding paragraph explicitly asks this question for integers r ≥ 2. The formal answer is negative: in an elementary abelian 2-group, odd h ≥ 3 and a subset larger than half the group give restricted sumset A and ordinary sumset G. These identities hold for every subset in that cardinality range, so they determine both minima.