An extension of the Erdős‐Tetali theorem

Abstract

Given a sequence , let r𝒜,h(n) denote the number of ways n can be written as the sum of h elements of 𝒜. Fixing h ≥ 2, we show that if f is a suitable real function (namely: locally integrable, O‐regularly varying and of positive increase) satisfying urn:x-wiley:rsa:media:rsa20812:rsa20812-math-0002 then there must exist with for which r𝒜,h + ℓ(n) = Θ(f(n)h + ℓ/n) for all ℓ ≥ 0. Furthermore, for h = 2 this condition can be weakened to . The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.