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New estimates of double trigonometric sums with exponential functions

M. Z. Garaev·A. A. Karatsuba
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Abstract

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.

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M. Z. Garaev, A. A. Karatsuba. 2005-04-01. New estimates of double trigonometric sums with exponential functions. https://arxiv.org/abs/math/0504026

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