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Averages of short exponential sums. (English) Zbl 0936.11048

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)

MSC:

11L07 Estimates on exponential sums
11J71 Distribution modulo one
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