Additive bases in infinite abelian semigroups, I

Abstract: An additive basis $A$ of a semigroup $T$ is a subset such that every element of $T$, up to a finite set of exceptions, may be written as a sum of one and the same number $h$ of elements from the basis. The minimal such number $h$ is called the order of the basis. We study bases in a class of infinite abelian semigroups, which we term translatable semigroups. These include all infinite abelian groups as well as the semigroup of nonnegative integers. We analyze the ``robustness" of bases. Such discussions have a long history in the semigroup ${\mathbf N}$, originating in the work of Erd\H os and Graham, continued by Deschamps and Farhi, Nathanson and Nash, Hegarty.... Thus we consider essential subsets of a basis $A$, that is, finite sets $F$ such that $A \setminus F$ is no longer a basis, and which are minimal. We show that any basis has only finitely many essential subsets, and we bound the number of essential subsets of cardinality $k$ of a basis of order $h$ in terms of $h$ and $k$.

Joint work with Benjamin Girard and Thai Hoang Lˆe.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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