A construction of maximal asymptotic nonbases
Deng-Rong Ling · International Journal of Number Theory · 2017
Let [Formula: see text] denote the set of all nonnegative integers and [Formula: see text] be a subset of [Formula: see text]. The set [Formula: see text] is called an asymptotic basis of order [Formula: see text] if every sufficiently large integer can be written as the sum of two elements of [Formula: see text]. Otherwise, [Formula: see text] is called an asymptotic nonbasis of order [Formula: see text]. Let [Formula: see text] denote the number of representations of [Formula: see text] in the form [Formula: see text], where [Formula: see text] and [Formula: see text]. An asymptotic nonbasis [Formula: see text] of order [Formula: see text] is called a maximal asymptotic nonbasis of order [Formula: see text] if [Formula: see text] is an asymptotic basis of order [Formula: see text] for every [Formula: see text]. In this paper, a maximal asymptotic nonbasis [Formula: see text] is constructed satisfying [Formula: see text] for all [Formula: see text] and [Formula: see text] as [Formula: see text], where [Formula: see text] is an increasing sequence of [Formula: see text].