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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 ${ℤ}_{2}$, ${ℤ}_{4}$, ${ℤ}_{7}$, ${ℤ}_{2}^{2}$, ${ℤ}_{2}^{4}$, or ${ℤ}_{2}^{2}×{ℤ}_{4}$.

##### Keywords
Abelian group, sumset, restricted addition, basis, $B_h$ set
##### Mathematical Subject Classification
Primary: 11B13
Secondary: 05B10, 11B75, 11P70, 20K01