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.

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

methods

On Minimal Asymptotic Basis of Order 4

A new class of minimal asymptotic bases

On Minimal Asymptotic Bases

In diesem Paragraphen sollen die in § l definierten xS-Basen genauer untersucht werden. Satz 1. Eine Zahlenmenge 9l ist dann und nur dann eine S-Basis, wenn es eine natürliche Zahl m und eine unendliche arithmetische Progression s, s + i, s + 2i, . . . mit s > 0 und t > 0 so gibt, daß jede Zahl dieser Progression Summe von höchstens m Zahlen aus 9l ist. Beweis. Wenn 9l die Voraussetzung erfüllt, so ist jede durch s t teilbare natürliche Zahl eine Summe von höchstens t m Summanden aus 9l, denn es ist

Additional excerpts

Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I.

Minimal bases and maximal nonbases in additive number theory

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background

A simple construction of minimal asymptotic bases

background or methods

Minimal bases and powers of 2

Jia and Nathanson [2] give a simple and explicit construction of minimal asymptotic bases of orderh for everyh≥2. They constructed minimal asymptotic bases from partition of N by means of powers of 2. In this paper, we extend the results of that paper to asymptotic bases constructed from partitions of N by means ofg-adic representations forg≥2. Corollary 3 shows that given partition N=W0⌣W1⌣...⌣Wh−1 such that eachWi contains infinitely many pairs of consecutive integers we can construct a minimal asymptotic bases of orderh in infinitely many way.