Zaharescu, Alexandru Averages of short exponential sums. (English) Zbl 0936.11048 Acta Arith. 88, No. 3, 223-231 (1999). Let \(M, N, P\) be positive integers satisfying \(N \leq P\) and let \(r(X)=f(X)/g(X)\) be a rational function, where \(f(X), g(X) \in {\mathbb{Z}}[X]\) and \(\deg f, \deg g \leq K_1\) for some fixed positive number \(K_1\). Assume that \(r(X)\) is not a polynomial and that all coefficients of \(r(X)\) are bounded by \(P^{K_2}\) for some constant \(K_2\). The author presents an upper bound for the second moment \[ \sum_{P \leq p \leq 2P} \sum_{m=1}^M\;\left |\mathop{{\sum}'}_{1 \leq n \leq N} e \left ( \frac{m r(n)}{p} \right)\right |^2, \] where \(\sum'\) means that the poles of \(r(X)\) are excepted. One application is concerned with the uniform distribution in \([0,1]\) of almost all sequences of fractional parts of type \(\{m r(n)/p\}_{1 \leq n \leq N}\). Reviewer: J.Hinz (Marburg) Cited in 1 Review MSC: 11L07 Estimates on exponential sums 11J71 Distribution modulo one Keywords:short exponential sums; upper bound for second moment; uniform distribution; sequences of fractional parts PDFBibTeX XMLCite \textit{A. Zaharescu}, Acta Arith. 88, No. 3, 223--231 (1999; Zbl 0936.11048) Full Text: DOI EuDML