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On the size of the maximum of incomplete Kloosterman sums

Part of: (Co)homology theory Limit theorems Exponential sums and character sums Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  15 April 2021

DANTE BONOLIS
DANTE BONOLIS*
Affiliation:
IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria. e-mails: [email protected]
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Abstract

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Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$. A classical problem in analytic number theory is bounding the maximum

$$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$
of the absolute value of the incomplete sums $(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $. In this very general context one of the most important results is the Pólya–Vinogradov bound
$$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$
where $\hat t:{\mathbb F_p} \to \mathbb C$ is the normalized Fourier transform of t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any $\varepsilon > 0$ there exists a large subset of $a \in \mathbb F_p^ \times $ such that for $${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$ we have
$$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$
as $p \to \infty $. Finally, we prove a result on the growth of the moments of ${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$.

MSC classification

Primary: 11L03: Trigonometric and exponential sums, general 11T23: Exponential sums
Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 60F10: Large deviations
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 563 - 590
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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