TY - JOUR

T1 - Inverse problems for minimal complements and maximal supplements

AU - Alon, Noga

AU - Kravitz, Noah

AU - Larson, Matt

N1 - Funding Information:
The first author is supported in part by NSF grant DMS-1855464 , ISF grant 281/17 , and BSF grant 2018267 . We thank the referee for several helpful comments.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/6

Y1 - 2021/6

N2 - 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.

AB - 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.

KW - Additive combinatorics

KW - Minimal complement

KW - Probabilistic combinatorics

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U2 - 10.1016/j.jnt.2020.10.009

DO - 10.1016/j.jnt.2020.10.009

M3 - Article

AN - SCOPUS:85098529535

VL - 223

SP - 307

EP - 324

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -