Green’s problem on additive complements of the squares

Abstract

Let A and B be two subsets of the nonnegative integers. We call A and B additive complements if all sufficiently large integers n can be written as a +b, where a ∈ A and b ∈ B . Let S = {12,22,32, · · ·} be the set of all square numbers. Ben Green was interested in the additive complement of S. He asked whether there is an additive complement B = {bn }n=1 ⊆Nwhich satisfies bn = π 2 16 n 2+o(n2). Recently, Chen and Fang proved that if B is such an additive complement, then limsup n→∞ π2 16 n 2 −bn n1/2 logn ≥ √ 2 π 1 log4 . They further conjectured that limsup n→∞ π2 16 n 2 −bn n1/2 logn =+∞. In this paper, we confirm this conjecture by giving a much more stronger result, i.e., limsup n→∞ π2 16 n 2 −bn n ≥ π 4 . 2020 Mathematics Subject Classification. 11B13, 11B75. Manuscript received 3rd August 2020, revised 19th August 2020, accepted 20th August 2020.