Equidistribution of continued fraction convergents in $ \mathrm{SL}(2,\mathbb{Z}_m) $ with an application to local discrepancy

Bence Borda

doi:10.3934/jmd.2025005

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2025, Volume 21: 327-359. Doi: 10.3934/jmd.2025005

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Equidistribution of continued fraction convergents in $ \mathrm{SL}(2,\mathbb{Z}_m) $ with an application to local discrepancy

Bence Borda

Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

Received: March 15, 2023

Revised: October 15, 2024

Published online: April 10, 2025

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Abstract

Consider the sequence of continued fraction convergents $ p_n/q_n $ to a random irrational number. We study the distribution of the sequences $ p_n \pmod{m} $ and $ q_n \pmod{m} $ with a fixed modulus $ m $, and more generally, the distribution of the $ 2 \times 2 $ matrix with entries $ p_{n-1}, p_n, q_{n-1}, q_n \pmod{m} $. Improving the strong law of large numbers due to Szüsz, Moeckel, Jager and Liardet, we establish the central limit theorem and the law of the iterated logarithm, as well as the weak and the almost sure invariance principles. As an application, we find the limit distribution of the maximum and the minimum of the Birkhoff sum for the irrational rotation with the indicator of an interval as test function. We also compute the normalizing constant in a classical limit law for the same Birkhoff sum due to Kesten, and dispel a misconception about its dependence on the test interval.

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Mathematics Subject Classification: Primary: 11K50, 37E10; Secondary: 37A50.

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References

[1]	J. Aaronson and M. Keane, The visits to zero of some deterministic random walks, Proc. London Math. Soc., 44 (1982), 535-553. doi: 10.1112/plms/s3-44.3.535.
[2]	G. Aggarwal and A. Ghosh, A generalized Lévy-Khintchine theorem, preprint, arXiv: 2408.15683, 2024.
[3]	G. Aggarwal and A. Ghosh, Counting and joint equidistribution of approximates, preprint, arXiv: 2401.02747, 2024.
[4]	G. Aggarwal and A. Ghosh, Two central limit theorems in Diophantine approximation, preprint, arXiv: 2306.02304, 2023.
[5]	M. Alam, A. Ghosh and S. Yu, Quantitative Diophantine approximation with congruence conditions, J. Théor. Nombres Bordeaux, 33 (2021), 261-271. doi: 10.5802/jtnb.1161.
[6]	A. Avila, D. Dolgopyat, E. Duryev and O. Sarig, The visits to zero of a random walk driven by an irrational rotation, Israel J. Math., 207 (2015), 653-717. doi: 10.1007/s11856-015-1186-4.
[7]	J. Beck, Probabilistic Diophantine Approximation. Randomness in Lattice Point Counting, Springer Monogr. Math., Springer, Cham, 2014. doi: 10.1007/978-3-319-10741-7.
[8]	P. Billingsley, Convergence of Probability Measures, second edition, John Wiley & Sons, Inc., New York, 1999.
[9]	B. Borda, Limit laws of maximal Birkhoff sums for circle rotations via quantum modular forms, Int. Math. Res. Not. IMRN, 2023 (2023), 19340-19389. doi: 10.1093/imrn/rnad107.
[10]	R. Bradley, Basic properties of strong mixing conditions. A survey and some open questions, Update of, and a supplement to, the 1986 original, Probab. Surv., 2 (2005), 107–144. doi: 10.1214/154957805100000104.
[11]	M. Bromberg and C. Ulcigrai, A temporal central limit theorem for real-valued cocycles over rotations, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 2304-2334. doi: 10.1214/17-AIHP872.
[12]	K. Dajani and C. Kraaikamp, A note on the approximation by continued fractions under an extra condition, New York J. Math., 3A (1997/98), 69-80.
[13]	D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations I. Convex bodies, Geom. Funct. Anal., 24 (2014), 85-115. doi: 10.1007/s00039-014-0254-y.
[14]	D. Dolgopyat and B. Fayad, Deviations of ergodic sums for toral translations II. Boxes, Publ. Math. Inst. Hautes Études Sci., 132 (2020), 293-352. doi: 10.1007/s10240-020-00120-2.
[15]	D. Dolgopyat and B. Fayad, Limit theorems for toral translations, Hyperbolic Dynamics, Fluctuations and Large Deviations, 227–277, Proc. Sympos. Pure Math., vol. 89, Amer. Math. Soc., Providence, RI, 2015. doi: 10.1090/pspum/089/01492.
[16]	A. Fisher and T. Schmidt, Distribution of approximants and geodesic flows, Ergodic Theory Dynam. Systems, 34 (2014), 1832-1848. doi: 10.1017/etds.2013.23.
[17]	L. Heinrich, Rates of convergence in stable limit theorems for sums of exponentially $\psi$-mixing random variables with an application to metric theory of continued fractions, Math. Nachr., 131 (1987), 149-165. doi: 10.1002/mana.19871310114.
[18]	M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Math. Appl., 547, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9940-5.
[19]	H. Jager and P. Liardet, Distributions arithmétiques des dénominateurs de convergents de fractions continues, Nederl. Akad. Wetensch. Indag. Math., 50 (1988), 181-197. doi: 10.1016/S1385-7258(88)80026-X.
[20]	H. Kesten, The discrepancy of random sequences $\{ kx \}$, Acta Arith., 10 (1964/65), 183-213. doi: 10.4064/aa-10-2-183-213.
[21]	H. Kesten, Uniform distribution mod 1, Ann. of Math. (2), 71 (1960), 445-471. doi: 10.2307/1969938.
[22]	H. Kesten, Uniform distribution mod 1. II, Acta Arith., 7 (1961/1962), 355-380. doi: 10.4064/aa-7-4-355-380.
[23]	C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239-301. doi: 10.2307/2118636.
[24]	R. Moeckel, Geodesics on modular surfaces and continued fractions, Ergodic Theory Dynam. Systems, 2 (1982), 69-83. doi: 10.1017/S0143385700009585.
[25]	F. Móricz, Moment inequalities and the strong law of large numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 35 (1976), 299-314. doi: 10.1007/BF00532956.
[26]	E. Nesharim, R. Rühr and R. Shi, Metric Diophantine approximation with congruence conditions, Int. J. Number Theory, 16 (2020), 1923-1933. doi: 10.1142/S1793042120500980.
[27]	M. Peligrad, Invariance principles for mixing sequences of random variables, Ann. Probab., 10 (1982), 968-981. doi: 10.1214/aop/1176993718.
[28]	W. Philipp and W. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc., vol. 2, no. 161, 1975. doi: 10.1090/memo/0161.
[29]	E. Rio, Asymptotic Theory of Weakly Dependent Random Processes, translated from the 2000 French edition, Probab. Theory Stoch. Model., vol. 80, Springer, Berlin, 2017.
[30]	L. Roçadas and J. Schoißengeier, On the local discrepancy of $(n\alpha)$-sequences, J. Number Theory, 131 (2011), 1492-1497. doi: 10.1016/j.jnt.2011.01.016.
[31]	J. Schoissengeier, A metrical result on the discrepancy of $(n \alpha)$, Glasgow Math. J., 40 (1998), 393-425. doi: 10.1017/S0017089500032742.
[32]	U. Shapira and B. Weiss, Geometric and arithmetic aspects of approximation vectors, preprint, arXiv: 2206.05329, 2024.
[33]	P. Szüsz, Verallgemeinerung und Anwendungen eines Kusminschen Satzes, Acta Arith., 7 (1961/62), 149-160. doi: 10.4064/aa-7-2-149-160.