On the additive completion of polynomial sets of integers

Abstract

Let f( x) = Σ t=0 n c t x t be a polynomial of degree n with nonnegative integral coefficients, and let U : a 1 < a 2 < … < a k be the set of integers of the form a i = f( i) that belong to the interval [1, N]. Let B : b 1 < b 2 < … < b l be an additive completion of U for the interval [1, N]. Then given ϵ > 0, we have kl > (1 + (n − 1) 2n 2 − ϵ) N for sufficiently large N. A similar result is also proved under more general conditions, which suffice for the verification of Hanani's conjecture.