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On character sums with distances on the upper half plane over a finite field. (English) Zbl 1253.11109
Let \(q\) be odd, and \(\alpha\) be a non-square in \({\mathbb F}_q\). Let \({\mathcal H}_q = \left\{ x + y \sqrt{\alpha} : x \in {\mathbb F}_q, y \in {\mathbb F}_q^* \right\}\) be upper half plane over the finite field \({\mathbb F}_q\). Let \(\mathcal E, \mathcal F\) be subsets of \({\mathcal H}_q\) with size \(E, F\), respectively, and let \(\psi\) be a non-trivial additive character on \({\mathbb F}_q\). The authors prove the estimate \[ \left| \sum_{w \in \mathcal E, z \in \mathcal F} \psi(\delta(w,z)) \right| \leq \min \left\{ q + \sqrt{2qE}, \sqrt{3} q^{5/4} \right\} \sqrt{EF}, \] where \(\delta\) is the distance function on \({\mathcal H}_q\). This has consequences for the Erdös distance problem on \({\mathcal H}_q\). The method of proof uses the techniques in [D. Hart, A. Iosevich, D. Koh and M. Rudnev [Trans. Am. Math. Soc. 363, No. 6, 3255–3275 (2011; Zbl 1244.11013)] and Le Anh Vinh [Electron. J. Comb. 15, No. 1, Research Paper R5, 18 p. (2008; Zbl 1206.05054)] in combination to produce a stronger result than otherwise obtainable independently.

MSC:
11T24 Other character sums and Gauss sums
11T60 Finite upper half-planes
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