Some extremal properties of trigonometric sums (Q1898273)

The author studies power sums of the type \(S_n (k\alpha)\), where \(S_n (\alpha)= \sum_{j=1}^n e(\alpha_j)\). For \(S\subset\mathbb{N}\) we denote \(V_n (\{S \}, \alpha)= \max\{ |S_n (k\alpha)|\): \(k\in S\}\), \(U_n (\{S\})= \min\{ V_n (\{S \}, \alpha)\): \(\alpha\in R_n\}\) and \(U_n (T)= U_n (\{1, 2, \dots, T\})\). The following main results are obtained: Theorem 1. For any \(a\in (0, 1)\) there exists a constant \(c_1 (a)\) such that \[ U_n (an^2)\leq c_1 (a) \sqrt {n}. \] Theorem 2. Let \(2n\leq T\leq e^{2n}\). Then \[ U_n (T)\geq c_2 \sqrt {n\log T/ \log (e^{2n} /\log T)} \] for some constant \(c_2\).