Essentialities in additive bases

Abstract

Let A A be an asymptotic basis for N 0 \mathbb {N}_0 of some order. By an essentiality of A A one means a subset P P such that A ∖ P A \backslash P is no longer an asymptotic basis of any order and such that P P is minimal among all subsets of A A with this property. A finite essentiality of A A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions: (i) Does every asymptotic basis of N 0 \mathbb {N}_0 possess some essentiality? (ii) Is the number of essential subsets of size at most k k of an asymptotic basis of order h h (a number they showed to be always finite) bounded by a function of k k and h 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 ≥ 2 h \geq 2 , of an asymptotic basis of order h h with no essentialities.