Geodesic covers and Erdős distinct distances in hyperbolic surfaces

Lu, Zhipeng; Meng, Xianchang

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

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 $\ge c\left(Y\right)N/logN$ for distinct distances determined by any $N$ points in $Y$, where $c\left(Y\right)>0$ is some constant depending only on $Y$. In particular, for $Y$ being modular surface or standard regular of genus $g\ge 2$, we evaluate $c\left(Y\right)$ explicitly in terms of $g$.

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

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

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

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

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

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

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

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

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

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

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

[11] Hart, Derrick; Iosevich, Alex; Koh, Doowon; Rudnev, Misha 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] Huber, H. Über eine neue Klass automorpher Functionen und eine Gitterpunktproblem in der hyperbolische Ebene, Comment. Math. Helv., Volume 30 (1956), pp. 20-62 | DOI

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

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

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

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

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

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

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

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

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

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

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

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

[25] Rudnev, Misha; Selig, J. M. 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] Selberg, Atle Göttingen lecture, Collected Works I, Springer, 1989

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

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

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