On the restricted sumsets containing powers of an integer

• Published in: European journal of combinatorics Vol. 114; p. 103795
• Main Authors: Yu, Wang-Xing, Chen, Yong-Gao, Chen, Shi-Qiang
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2023
• ISSN: 0195-6698; 1095-9971
• DOI: 10.1016/j.ejc.2023.103795

In 2008, Lev proved that for any sufficiently large positive integer n, if A⊆[1,n] with gcdA=1 and |A|>6n/19, then there is a power of 2 that can be represented as the sum of at most 5 distinct elements of A. In this paper, as a corollary of our main results, we prove that for any ɛ>0 and any sufficiently large positive integer n, if A⊆[1,n] with gcdA=1 and |A|>(1/6+ɛ)n, then there is a power of 2 that can be represented as the sum of at most 36 distinct elements of A. On the other hand, there are infinitely many positive integers n, for each of which there exists B⊆[1,n] with gcdB=1 and |B|>n/6 such that no power of 2 can be represented as the sum of distinct elements of B.