Essentialities in additive bases

TL;DR

This paper investigates the concept of essentialities in additive bases, providing a negative answer to whether all bases have essentialities and confirming that the number of small essential subsets is bounded by a function of their size and order.

Contribution

It proves that not all asymptotic bases possess essentialities and establishes a bound on the number of essential subsets of a given size for bases of a fixed order.

Findings

Not all asymptotic bases have essentialities.

The number of essential subsets of size at most k is bounded by a function of k and h.

Explicit constructions of bases without essentialities for any order h >= 2.

Abstract

Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\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 N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities.