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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
##### Keywords:
character sums; finite upper half plane
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##### References:
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