Geodesic covers and Erdős distinct distances in hyperbolic surfaces

Zhipeng Lu; Xianchang Meng

doi:10.5802/ambp.422

Zhipeng Lu ¹ ; Xianchang Meng ²

¹ Shenzhen MSU-BIT University & Guangdong Laboratory of Machine Perception and Intelligent Computing Shenzhen, Guangdong 518172, China
² School of Mathematics Shandong University, Jinan Shandong 250100, China

Annales mathématiques Blaise Pascal, Tome 30 (2023) no. 2, pp. 201-217.

Résumé

In this paper, we introduce the notion of “geodesic cover” for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic surface. Then we use estimates of geodesic-covering numbers to study the distinct distances problem in hyperbolic surfaces. Especially, for $Y$ from a large class of hyperbolic surfaces, we establish the nearly optimal bound $\geq c (Y) N / log N$ for distinct distances determined by any $N$ points in $Y$ , where $c (Y) > 0$ is some constant depending only on $Y$ . In particular, for $Y$ being modular surface or standard regular of genus $g \geq 2$ , we evaluate $c (Y)$ explicitly in terms of $g$ .

Publié le : 2024-04-30

DOI : 10.5802/ambp.422

Classification : 52C10, 11P21, 20H10
Mots clés : Erdős distinct distances, hyperbolic surface, hyperbolic circle problem, equilateral dimension

Affiliations des auteurs :

Zhipeng Lu ¹ ; Xianchang Meng ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Geodesic covers and {Erd\H{o}s} distinct distances in hyperbolic surfaces},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {201--217},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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     doi = {10.5802/ambp.422},
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Zhipeng Lu; Xianchang Meng. Geodesic covers and Erdős distinct distances in hyperbolic surfaces. Annales mathématiques Blaise Pascal, Tome 30 (2023) no. 2, pp. 201-217. doi : 10.5802/ambp.422. https://ambp.centre-mersenne.org/articles/10.5802/ambp.422/

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