A remark on perturbations of sine and cosine sums

Abstract

Consider a collection $\lambda_1<...<\lambda_N$ of distinct positive integers and the quantities $$ M_1 = M_1(\lambda_1,...,\lambda_N) = \max_{0\le x \le 2\pi} |\sum_{j=1}^N \sin{\lambda_j x}| $$ and $$ M_2 = M_2(\lambda_1,...,\lambda_N) = - \min_{0\le x \le 2\pi} \sum_{j=1} \cos{\lambda_j x}. $$ Prompted by a discussion with G. Benke we prove that collections of frequencies $\lambda_j$ which have $M_1 = o(N)$ or $M_2 = o(N)$ are unstable, in the sense that one can perturb the $\lambda_j$ by one each and get $M_1 \ge c N$ and $M_2 \ge c N$.