On functional properties of incomplete Gaussian sums. (English) Zbl 0728.11039

Incomplete Gaussian sums are exponential sums \(\sum_{n\in \omega}\exp (2\pi ian^ 2/q)\), where \((a,q)=1\) and \(\omega\) is an interval of length at most q. These can be expressed in terms of the function \[ H(x_ 1,x_ 2)=\sum_{n\neq 0}\frac{\exp (2\pi i(n^ 2x_ 2+nx_ 1))}{2\pi in}, \] where the sum of the series is understood as the limit of the symmetric partial sums. By an analysis of this series, it is proved, for instance, that if \(\omega\) runs over a system of nonoverlapping intervals in the interval [0,q], then for each positive \(\epsilon >0\), the number of intervals such that the modulus of the corresponding incomplete Gaussian sums exceeds \(\epsilon\sqrt{q}\) is at most \(c\epsilon^{-2}\), where c is an absolute positive constant. Also, the size of the implied constant in the classical estimate \(\ll \sqrt{q}\) for incomplete Gaussian sums is discussed.
Reviewer: M.Jutila (Turku)


11L05 Gauss and Kloosterman sums; generalizations
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