On complete sequences

摘要

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence of positive integers and a positive real number alpha, let S (alpha) denote the sequence , where denotes the greatest integer not greater than x. Let . Hegyvari [6] proved that if , for all integers , where , and , then , where is the Lebesgue measure of U (S) . Yong-Gao Chen and the first author [4] proved that, if for all integers , where , then . In this paper, we prove that the conclusion holds for 1 < gamma <= 4 root 13 = 1.898....