On perfect bases in finite Abelian groups

Bajnok, Béla; Berson, Connor; Just, Hoang Anh

doi:10.2140/involve.2022.15.525

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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

DOI: 10.2140/involve.2022.15.525

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 $ℤ_{2}$ , $ℤ_{4}$ , $ℤ_{7}$ , $ℤ_{2}^{2}$ , $ℤ_{2}^{4}$ , or $ℤ_{2}^{2} \times ℤ_{4}$ .

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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

Vol. 15, No. 3, 2022