Dense minimal asymptotic bases of order two

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• 作者：Miroslawa Jań ; czak ; Tomasz Schoen
• 关键词：Asymptotic bases ; Additive number theory
• 刊名：Journal of Number Theory
• 出版年：2010
• 出版时间：March 2010
• 年：2010
• 卷：130
• 期：3
• 页码：580-585
• 全文大小：126 K

文摘

We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erdős and Nathanson showed that for every h2 there exists a minimal asymptotic basis A of order h with d(A)=1/h, where d(A) denotes the density of A. Erdős and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order h2 such that A(k)=k/h+O(1). Moreover, they asked if there exists a minimal asymptotic basis with lim sup(a_i+1−a_i)=3. In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition