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

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 = {Zhipeng Lu and Xianchang Meng},
     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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     number = {2},
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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/

Bibliographie
Cité par

[1] Noga Alon; Vitali D. Milman Embedding of $ℓ_{\infty}^{k}$ in finite-dimensional Banach spaces, Isr. J. Math., Volume 45 (1983), pp. 265-280 | DOI | MR | Zbl

[2] Florin P. Boca; Alexandru A. Popa; Alexandru Zaharescu Pair correlation of hyperbolic lattice angles, Int. J. Number Theory, Volume 10 (2014) no. 8, pp. 1955-1989 | DOI | MR

[3] Jean Bourgain; Netz Katz; Terence Tao A sum-product estimate in finite fields, and applications, Geom. Funct. Anal., Volume 14 (2004) no. 1, pp. 27-57 | DOI | MR | Zbl

[4] Jean Delsarte Sur le Gitter Fuchsien, C. R. Acad. Sci. Paris, Volume 214 (1942), pp. 147-149 | MR | Zbl

[5] Pál Erdős On sets of distances of $n$ points, Am. Math. Mon., Volume 53 (1946), pp. 248-250 | DOI | MR | Zbl

[6] Kenneth J. Falconer On the Hausdorff dimensions of distance sets, Mathematika, Volume 32 (1985), pp. 206-212 | DOI | MR | Zbl

[7] Larry Guth Polynomial Methods in Combinatorics, University Lecture Series, 64, American Mathematical Society, 2016 | DOI

[8] Larry Guth; Alex Iosevich; Yumeng Ou; Hong Wang On Falconer’s distance set problem in the plane, Invent. Math., Volume 219 (2020) no. 3, pp. 779-830 | DOI | MR

[9] Larry Guth; Nets Hawk Katz On the Erdős distinct distances problem in the plane, Ann. Math., Volume 181 (2015) no. 1, pp. 155-190 | DOI

[10] Richard K. Guy An Olla-Podrida of open problems, often oddly posed, Am. Math. Mon., Volume 90 (1983), pp. 196-200 | MR

[11] Derrick Hart; Alex Iosevich; Doowon Koh; Misha Rudnev Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture, Trans. Am. Math. Soc., Volume 363 (2011) no. 6, pp. 3255-3275 | DOI | Zbl

[12] H. Huber Über eine neue Klass automorpher Functionen und eine Gitterpunktproblem in der hyperbolische Ebene, Comment. Math. Helv., Volume 30 (1956), pp. 20-62 | DOI

[13] H. Huber Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann., Volume 138 (1959), pp. 1-26 | DOI | MR

[14] Alex Iosevich What is ... Falconer’s conjecture?, Notices Am. Math. Soc., Volume 66 (2019), pp. 552-555 | MR

[15] Alex Iosevich; Misha Rudnev Erdős distance problem in vector spaces over finite fields, Trans. Am. Math. Soc., Volume 359 (2007), pp. 6127-6142 | DOI | Zbl

[16] Henryk Iwaniec Spectral methods of automorphic forms, Graduate Studies in Mathematics, 53, American Mathematical Society, 2002

[17] Svetlan Katok Fuchsian Groups, University of Chicago Press, 1992

[18] Dubi Kelmer; Alex Kontorovich On the pair correlation density for hyperbolic angles, Duke Math. J., Volume 164 (2015) no. 3, pp. 473-509 | MR

[19] Jack Koolen; Monique Laurent; Alexander Schrijver Equilateral dimension of the rectilinear space, Des. Codes Cryptography, Volume 21 (2000) no. 1-3, pp. 149-164 | DOI | MR

[20] Gregory Margulis Applications of ergodic theory to the investigation of manifolds of negative curvature, Funct. Anal. Appl., Volume 4 (1969), pp. 335-336

[21] Xianchang Meng Distinct distances on hyperbolic surfaces, Trans. Am. Math. Soc., Volume 375 (2022) no. 3, pp. 3713-3731 | MR

[22] János Pach; Micha Sharir On the number of incidences between points and curves, Comb. Probab. Comput., Volume 7 (1998) no. 1, pp. 121-127 | DOI | MR

[23] Samuel J. Patterson A lattice point problem in hyperbolic space, Mathematika, Lond., Volume 22 (1974), pp. 81-88 | DOI | MR | Zbl

[24] Ralph Phillips; Zeév Rudnick The circle problem in the hyperbolic plane, J. Funct. Anal., Volume 121 (1994) no. 1, pp. 78-116 | DOI | MR

[25] Misha Rudnev; J. M. Selig On the use of the Klein quadric for geometric incidence problems in two dimensions, SIAM J. Discrete Math., Volume 30 (2016) no. 2, pp. 934-954 | DOI | MR

[26] Atle Selberg Göttingen lecture, Collected Works I, Springer, 1989

[27] Adam Sheffer Distinct distances: open problems and current bounds (2014) | arXiv

[28] Adam Sheffer; Joshua Zahl Distinct distances in the complex plane, Trans. Am. Math. Soc., Volume 374 (2021) no. 9, pp. 6691-6725 | DOI | MR

[29] József Solymosi; Van H. Vu Near optimal bounds for the Erdős distinct distances problem in high dimensions, Combinatorica, Volume 28 (2008) no. 1, pp. 113-125 | DOI | Zbl

Cité par Sources :