On a problem of Erdos, Nathanson and Sarkozy
On a problem of Erdos, Nathanson and Sarkozy
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关于鄂尔多斯、内桑森和萨科齐的问题
DOI：
10.1016/j.jnt.2019.02.025
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发表时间：
2019
影响因子：
0.7
通讯作者：
Li Ya-Li
中科院分区：
文献类型：
--
作者：
Chen Yong-Gao;Li Ya-Li
文摘1988年文本,Erdő年代,内桑森和萨科齐证明,如果是一组与低密度的渐近非负整数1 / k, k是一个正整数,然后(k + 1)必须包含一个无限等差数列与差异最多2−k, k (k + 1)一个是k + 1的所有元素的集合,并问(k + 1)必须包含一个无限等差数列与差异最多O (k)。本文证明了对于每一个足够大的整数k，存在一个密度为1/k的非负整数集a，使得（k+ 1） a不包含差值小于k1.5的无穷等差数列，从而解决了这个问题。我们还研究了当A具有较低的渐近密度0< w< 1时的类似问题。有关本文的视频摘要，请访问https://youtu。/ B1QJc2IFSIw。
Abstract Text In 1988, Erdős, Nathanson and Sárközy proved that, if A is a set of nonnegative integers with the lower asymptotic density 1/k, where k is a positive integer, then (k+ 1) A must contain an infinite arithmetic progression with difference at most k 2− k, where (k+ 1) A is the set of all sums of k+ 1 elements of A, and asked if (k+ 1) A must contain an infinite arithmetic progression with difference at most O (k). In this paper, we solve this problem by proving that, for every sufficiently large integer k, there exists a set A of nonnegative integers with the lower asymptotic density 1/k such that (k+ 1) A does not contain an infinite arithmetic progression with difference less than k 1.5. We also study the similar problem when A has the lower asymptotic density 0< w< 1. Video For a video summary of this paper, please visit https://youtu. be/B1QJc2IFSIw.