On irregularities of distribution in shifts and dilations of integer sequences. I
Authors' review: The goal of this series is to study, for any sequence of complex numbers \(\varepsilon_ 1,...,\varepsilon_ N\) and any sequence of positive integers \(b_ 1,b_ 2,...\), the quantity \(\max | \sum^{t}_{j=1}\epsilon_{a+b_ jq}|\) where the maximum is taken over all integers \(a\) and all positive integers \(q\) and \(t\). In the above sum we take only those terms \(\varepsilon_{a+b_ jq}\) for which \(1\leq a+b_ jq\leq N\). For example, let \(N, L\) and \(b_ 1,...,b_ t\) be positive integers with \(b_ 1<b_ 2<...<b_ t\leq L\) and \(L\leq N^{1/2}\). Let \(\varepsilon_ 1,...,\varepsilon_ N\) be complex numbers of absolute value at least one. We prove that \[ \max_{1\leq q\leq 14L;\quad -14L^ 2\leq m\leq N}| \sum^{t}_{j=1}\epsilon_{m+b_ jq}| \geq t/(30 L^{1/2}). \] Further, let \(\delta\) be a positive real number, and let \(\varepsilon_ 1,...,\varepsilon_ N\) be complex numbers of absolute value at least one. We show that there exists a number \(c(\delta)\) which is effectively computable in terms of \(\delta\) such that if \(N>c(\delta)\) then \[ \max_{1\leq q,t\leq N^{1/3};\quad -N\leq m\leq N}| \sum^{t}_{x=1}\epsilon_{m+x^ 2q}| >N^{1/6}\quad \exp (- (1+\delta)(\log 2\log N)/3\log \log N). \] We apply results of the above type to give lower bounds for character sums.
scientific article; zbMATH DE number 3948404