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September 2020 Kloosterman sums with twice-differentiable functions
Igor E. Shparlinski, Marc Technau
Funct. Approx. Comment. Math. 63(1): 113-124 (September 2020). DOI: 10.7169/facm/1845

Abstract

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \floor{f(n)}+ y \floor{f(n)}^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $\floor{f(n)}^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\floor{f(n)}^{-1} \pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ \floor{t^c}$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.

Citation

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Igor E. Shparlinski. Marc Technau. "Kloosterman sums with twice-differentiable functions." Funct. Approx. Comment. Math. 63 (1) 113 - 124, September 2020. https://doi.org/10.7169/facm/1845

Information

Published: September 2020
First available in Project Euclid: 14 December 2019

MathSciNet: MR4149513
Digital Object Identifier: 10.7169/facm/1845

Subjects:
Primary: 11B83, 11L07
Secondary: 11L05

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.63 • No. 1 • September 2020
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