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Pré-Publication, Document De Travail Année : 2020

On additive bases in infinite abelian semigroups

Résumé

In this paper, building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in a class of infinite abelian semigroups, which we term \em translatable \em semigroups. These include all numerical semigroups as well as all infinite abelian groups. We show that, for every such semigroup $T$, the number of essential subsets of any additive basis is finite, and also that the number $E_T(h, k)$ of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon ${\mathbf{N}}$. Also, using invariant means, we address a classical problem, initiated by Erd\H{o}s and Graham and then generalized by Nash and Nathanson both in the case of ${\mathbf{N}}$, of estimating the maximal order $X_T(h, k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that, whenever $T$ is a translatable semigroup, $X_T(h, k)$ is $O(h^{2k+1})$ for every integer $k \ge 1$. This result is new even in the case where $k = 1$ and $T$ is an infinite abelian group. Besides the maximal order $X_T(h, k)$, the typical order $S_T(h, k)$ is also studied.
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Dates et versions

hal-02470506 , version 1 (07-02-2020)
hal-02470506 , version 2 (12-01-2021)

Identifiants

  • HAL Id : hal-02470506 , version 1

Citer

Pierre-Yves Bienvenu, Benjamin Girard, Thái Hoàng Lê. On additive bases in infinite abelian semigroups. 2020. ⟨hal-02470506v1⟩
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