Seminar at SMU Delhi

February 4, 2020 (Tuesday) , 3:30 PM at Webinar
Speaker: Pierre-Yves Bienvenu, Camille Jordan Institute, Lyon, France
Title: Additive bases in infinite abelian semigroups
Abstract of Talk
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 \textit{translatable} semigroups. These include all infinite abelian groups as well as the semigroup $\N$ of nonnegative integers. 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 theorems extend the reach of two important results, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $\N$. Also, using invariant means, we address a classical problem, initiated by Erdös and Graham and extended by Nash and Nathanson both in the case of $\N$, of estimating the maximal order $X_T(h,k)$ that a basis of co-cardinality $k$ contained in a given additive basis of order $h$ 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 $k=1$ and $T$ is an infinite abelian group. Besides the maximal order $X_T(h,k)$, we also study the typical order $S_T(h,k)$. Joint work with Benjamin Girard and Thai Hoang Lê.