Geodesic
Geodesic covers and Erdős distinct distances in hyperbolic surfaces
Lu, Zhipeng1 ; Meng, Xianchang2
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

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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 ≥c(Y)N/logN 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≥2, we evaluate c(Y) explicitly in terms of g.

Publié le :
DOI : 10.5802/ambp.422
Classification : 52C10, 11P21, 20H10
Keywords: Erdős distinct distances, hyperbolic surface, hyperbolic circle problem, equilateral dimension

Lu, Zhipeng 1 ; Meng, Xianchang 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Lu, Zhipeng and Meng, Xianchang},
     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},
     volume = {30},
     number = {2},
     year = {2023},
     doi = {10.5802/ambp.422},
     language = {en},
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Lu, Zhipeng; Meng, Xianchang. 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

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