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On perfect bases in finite Abelian groups

Béla Bajnok, Connor Berson and Hoang Anh Just

Vol. 15 (2022), No. 3, 525–536
Abstract

Let G be a finite abelian group and s be a positive integer. A subset A of G is called a perfect s-basis of G if each element of G can be written uniquely as the sum of at most s (not necessarily distinct) elements of A; similarly, we say that A is a perfect restricted s-basis of G if each element of G can be written uniquely as the sum of at most s distinct elements of A. We prove that perfect s-bases exist only in the trivial cases of s = 1 or |A| = 1. The situation is different with restricted addition where perfection is more frequent; here we treat the case of s = 2 and prove that G has a perfect restricted 2-basis if, and only if, it is isomorphic to 2, 4, 7, 22, 24, or 22 × 4.

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Keywords
Abelian group, sumset, restricted addition, basis, $B_h$ set
Mathematical Subject Classification
Primary: 11B13
Secondary: 05B10, 11B75, 11P70, 20K01
Milestones
Received: 29 October 2021
Accepted: 31 December 2021
Published: 2 December 2022

Communicated by Steven J. Miller
Authors
Béla Bajnok
Department of Mathematics
Gettysburg College
Gettysburg, PA
United States
Connor Berson
Asymmetrik
Annapolis Junction, MD
United States
Hoang Anh Just
Department of Electrical and Computer Engineering
Virginia Tech
Blacksburg, VA
United States