Inverse problems for minimal complements and maximal supplements

Noga Alon, Noah Kravitz, Matt Larson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Given a subset W of an abelian group G, a subset C is called an additive complement for W if W+C=G; if, moreover, no proper subset of C has this property, then we say that C is a minimal complement for W. It is natural to ask which subsets C can arise as minimal complements for some W. We show that in a finite abelian group G, every non-empty subset C of size |C|≤22/3|G|1/3/((3elog⁡|G|)2/3 is a minimal complement for some W. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for “dual” problems about maximal supplements.

Original languageEnglish
Pages (from-to)307-324
Number of pages18
JournalJournal of Number Theory
StatePublished - Jun 2021


FundersFunder number
National Science FoundationDMS-1855464
Bonfils-Stanton Foundation2018267
Israel Science Foundation281/17


    • Additive combinatorics
    • Minimal complement
    • Probabilistic combinatorics


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