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Equidistribution of Farey sequences on horospheres in covers of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$ and applications

Part of: Diophantine equations Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory Sequences and sets

Published online by Cambridge University Press:  07 October 2019

BYRON HEERSINK
BYRON HEERSINK*
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH43210, USA email [email protected], [email protected]
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Abstract

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

Keywords

Farey sequenceshorosphereequidistributionDiophantine approximationFrobenius numbers

MSC classification

Primary: 37A17: Homogeneous flows 11B57: Farey sequences; the sequences ${1^k, 2^k, cdots}$
Secondary: 37A45: Relations with number theory and harmonic analysis 11K60: Diophantine approximation 11D07: The Frobenius problem
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 471 - 493
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Copyright
© Cambridge University Press, 2019

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References

Athreya, J. S. and Cheung, Y.. A Poincaré section for the horocycle flow on the space of lattices. Int. Math. Res. Not. IMRN 2014(10) (2014), 2643–2690.10.1093/imrn/rnt003CrossRefGoogle Scholar
Athreya, J. S. and Ghosh, A.. The Erdős–Szüsz–Turán distribution for equivariant processes. Enseign. Math. (2) 64 (2018), 1–21.Google Scholar
Bass, H., Lazard, M. and Serre, J.-P.. Sous-groupes d’indice fini dans SL(n, ℤ). Bull. Amer. Math. Soc. 70 (1964), 385–392.10.1090/S0002-9904-1964-11107-1CrossRefGoogle Scholar
Boca, F. P.. A problem of Erdős, Szüsz, and Turán concerning diophantine approximations. Int. J. Number Theory 4(4) (2008), 691–708.10.1142/S1793042108001626CrossRefGoogle Scholar
Boca, F. P., Cobeli, C. and Zaharescu, A.. A conjecture of R. R. Hall on Farey points. J. Reine Angew. Math. 535 (2001), 207–236.Google Scholar
Einsiedler, M., Mozes, S., Shah, N. and Shapira, U.. Equidistribution of primitive rational points on expanding horospheres. Compos. Math. 152(4) (2016), 667–692.CrossRefGoogle Scholar
Erdős, P., Szüsz, P. and Turán, P.. Remarks on the theory of Diophantine approximation. Colloq. Math. 6 (1958), 119–126.10.4064/cm-6-1-119-126CrossRefGoogle Scholar
Eskin, A. and McMullen, C.. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71(1) (1993), 181–209.10.1215/S0012-7094-93-07108-6CrossRefGoogle Scholar
Fisher, A. M. and Schmidt, T. A.. Distribution of approximants and geodesic flows. Ergod. Th. & Dynam. Sys. 34(6) (2014), 1832–1848.10.1017/etds.2013.23CrossRefGoogle Scholar
Heersink, B.. Poincaré sections for the horocycle flow in covers of SL(2, ℝ)/SL(2, ℤ) and applications to Farey fraction statistics. Monatsh. Math. 179(3) (2016), 389–420.10.1007/s00605-015-0873-xCrossRefGoogle Scholar
Kesten, H.. Some probabilistic theorems on Diophantine approximations. Trans. Amer. Math. Soc. 103 (1962), 189–217.10.1090/S0002-9947-1962-0137692-1CrossRefGoogle Scholar
Kesten, H. and Sós, V. T.. On two problems of Erdős, Szüsz and Turán concerning diophantine approximations. Acta Arith. 12 (1966–1967), 183–192.10.4064/aa-12-2-183-192CrossRefGoogle Scholar
Lee, M. and Marklof, J.. Effective equidistribution of rational points on expanding horospheres. Int. Math. Res. Not. IMRN 2018(21) (2018), 6581–6610.10.1093/imrn/rnx081CrossRefGoogle Scholar
Li, H.. Effective limit distribution of the Frobenius numbers. Compos. Math. 151(5) (2015), 898–916.10.1112/S0010437X14007866CrossRefGoogle Scholar
Marklof, J.. The asymptotic distribution of Frobenius numbers. Invent. Math. 181(1) (2010), 179–207.10.1007/s00222-010-0245-zCrossRefGoogle Scholar
Marklof, J.. Horospheres and Farey fractions. Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory (Contemporary Mathematics, 532) . American Mathematical Society, Providence, RI, 2010, pp. 97–106.10.1090/conm/532/10485CrossRefGoogle Scholar
Marklof, J.. Fine-scale statistics for the multidimensional Farey sequence. Limit Theorems in Probability, Statistics and Number Theory (Springer Proceedings in Mathematics and Statistics, 42) . Springer, Heidelberg, 2013, pp. 49–57.10.1007/978-3-642-36068-8_3CrossRefGoogle Scholar
Marklof, J. and Strömbergsson, A.. The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2) 172(3) (2010), 1949–2033.10.4007/annals.2010.172.1949CrossRefGoogle Scholar
Mennicke, J. L.. Finite factor groups of the unimodular group. Ann. of Math. (2) 81 (1965), 31–37.10.2307/1970380CrossRefGoogle Scholar
Ratner, M.. On Raghunathan’s measure conjecture. Ann. of Math. (2) 134(3) (1991), 545–607.10.2307/2944357CrossRefGoogle Scholar
Shah, N.. Limit distributions of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 106(2) (1996), 105–125.10.1007/BF02837164CrossRefGoogle Scholar
Xiong, M. and Zaharescu, A.. A problem of Erdős–Szüsz–Turán on diophantine approximation. Acta Arith. 125(2) (2006), 163–177.CrossRefGoogle Scholar