Additive complements of the squares (Q2741207)

Consider sets \(B\subset [0,N]\) of integers with the property that every integer \(1\leq n\leq N\) is of the form \(n=b+k^2\) with \(b\in B\), and put \(b(N)= \min |B|\) over such sets. By taking the first integers one sees easily that \(b(N)\leq 2\sqrt N+O(1)\), and one is tempted to conjecture that this is the correct order of magnitude. It is known that \(b(N) \geq (4/\pi +o(1)) \sqrt N\) (Cilleruelo and Habsieger). This paper proves a bound of type \(|B|\geq (f(\delta) + o(1)) \sqrt N\) under the additional assumption that \(B\subset [0, \delta N]\). Here \(f\) is a function such that \(f(1)=4/\pi \), so that we regain the Cilleruelo-Habsieger bound, and \(f(0)=2\), so this establishes the conjectured bound for sets in which all elements are small.