A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ not necessarily distinct elements of $A$ The asymptotic basis $A$ is minimal if removing any element of $A$ destroys every representation of infinitely many integers, and so $A\setminus \{a\}$ is not an asymptotic basis of order $h$ for all $a\in A$ In this paper, a new class of minimal asymptotic bases is constructed

A new class of minimal asymptotic bases

On Minimal Asymptotic Bases

Abstract Let N denote the set of all nonnegative integers and A be a subset of N. Let W be a nonempty subset of N. Denote by F∗(W ) the set of all finite, nonempty subsets of W . Fix integer g ≥ 2, let Ag(W ) be the set of all numbers of the form ∑ f∈F afg f where F ∈ F∗(W ) and 1 ≤ af ≤ g − 1. For i = 0, 1, 2, 3, let Wi = {n ∈ N | n ≡ i (mod 4)}. In this paper, we show that the set A = ∪3 i=0 Ag(Wi) is a minimal asymptotic basis of order four.

Li Jiawen

Papers

On Minimal Asymptotic Basis of Order 4