On additive representation functions

Abstract

Let 𝒜 = {a1 < a2 < a3 < ⋯ < an < ⋯} be an infinite sequence of nonnegative integers and let R2(n) = |{(i, j) : ai + aj = n; ai, aj ∈ 𝒜; i ≤ j}|. We define [Formula: see text]. We prove that if the L∞-norm of [Formula: see text] is small, then the L1-norm of [Formula: see text] is large.