A remark on perturbations of sine and cosine sums

Abstract

Consider a collection ${\lambda }_1<...<{\lambda }_N$ of distinct positive integers and the quantities $$M_1=M_1({\lambda }_1,...,{\lambda }_N)=\underset{0\le x\le 2\pi }{\max }|\sum _{j=1}^N\sin {\lambda }_{j}x|$$ and $$M_2=M_2({\lambda }_1,...,{\lambda }_N)=-\underset{0\le x\le 2\pi }{\min }\sum _{j=1}\cos {\lambda }_{j}x.$$ 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 $M_1\ge cN$ and $M_2\ge cN$.

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