A remark on perturbations of sine and cosine sums

Abstract

Consider a collection $λ_1<...<λ_N$ of distinct positive integers and the quantities $$ M_1 = M_1(λ_1,...,λ_N) = \max_{0\le x \le 2π} |\sum_{j=1}^N \sin{λ_j x}| $$ and $$ M_2 = M_2(λ_1,...,λ_N) = - \min_{0\le x \le 2π} \sum_{j=1} \cos{λ_j x}. $$ Prompted by a discussion with G. Benke we prove that collections of frequencies $λ_j$ which have $M_1 = o(N)$ or $M_2 = o(N)$ are unstable, in the sense that one can perturb the $λ_j$ by one each and get $M_1 \ge c N$ and $M_2 \ge c N$.