On the additive completion of polynomial sets (Q1804193)

Let \(P\) be a polynomial of degree \(k\geq 2\) with non-negative coefficients and let \(B\) be the set of integers such that every \(n\leq N\) can be written as the sum of an element from \(B\) and a \(P(\lambda)\) for some integer \(\lambda\). A conjecture of Erdős that when \(P(x)= x^ 2\), \(| B|> (1+\varepsilon) \sqrt{N}\) for some \(\varepsilon>0\) was proved by Moser with \(\varepsilon= 0.6\). The result was generalized to polynomials \(P\) by a number of people. The author improves the more general result to \[ | B| P^{-1} (N)> \Bigl( \bigl( 1- {\textstyle {1\over k}}\bigl)^{-1} {\textstyle {{\sin \pi k} \over {\pi k}}}- \varepsilon \Bigl)N, \] for \(N\) sufficiently large. This leads to improved constants in lower bound estimates for \(| B|\) when \(P(x)= x^ 2\) and \(P(x)= x^ 3\).