Essentialities in additive bases

Abstract:

Let $A$ be an asymptotic basis for $\mathbb {N}_0$ of some order. By an essentiality of $A$ one means a subset $P$ such that $A \backslash P$ is no longer an asymptotic basis of any order and such that $P$ is minimal among all subsets of $A$ with this property. A finite essentiality of $A$ is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions: (i) Does every asymptotic basis of $\mathbb {N}_0$ possess some essentiality? (ii) Is the number of essential subsets of size at most $k$ of an asymptotic basis of order $h$ (a number they showed to be always finite) bounded by a function of $k$ and $h$ only? We answer the latter question in the affirmative and answer the former in the negative by means of an explicit construction, for every integer $h \geq 2$, of an asymptotic basis of order $h$ with no essentialities.