On additive bases in infinite abelian semigroups

Journal of Combinatorial Algebra

article

Pierre-Yves Bienvenu¹  Benjamin Girard²  Thái Hoàng Lê³ 

[1] Trinity College Dublin;Sorbonne Université and Université de Paris;The University of Mississippi

关键词: Additive combinatorics;    infinite abelian group;    translatable semigroup;    Grothendieck group;    additive basis;    essential subset;    invariant mean;   

DOI  :  10.4171/jca/67

学科分类：外科医学

来源: European Mathematical Society

【 摘 要 】

Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group TTT, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality kkk contained in an additive basis of order at most hhh can be bounded in terms of hhh and kkk alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon N\mathbf{N}N. Also, using invariant means, we address a classical problem, initiated by Erdős and Graham and then generalized by Nash and Nathanson both in the case of N\mathbf{N}N, of estimating the maximal order XT(h,k)X_T(h,k)XT​(h,k) that a basis of cocardinality kkk contained in an additive basis of order at most hhh can have. Among other results, we prove that XT(h,k)=O(h2k+1)X_T(h,k)=O(h^{2k+1})XT​(h,k)=O(h2k+1) for every integer k≥1k \ge 1k≥1. This result is new even in the case where k=1k=1k=1. Besides the maximal order XT(h,k)X_T(h,k)XT​(h,k), the typical order ST(h,k)S_T(h,k)ST​(h,k) is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in N\mathbf{N}N and in abelian groups.

【 授权许可】

CC BY