Additive complements of the squares

Abstract

TextTwo infinite sequences A and B of nonnegative integers are called additive complements, if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S={12,22,…}. The following result is proved: if B={bn}n=1∞ with bn≥π216n2−0.57n12log⁡n−βn12 for all positive integers n and any given constant β, then B is not an additive complement of S. In particular, B={⌊π216n2⌋|n=1,2,…} is not an additive complement of S. VideoFor a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8.