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Poincaré sections for the horocycle flow in covers of \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\) and applications to Farey fraction statistics

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Abstract

For a given finite index subgroup \(H\subseteq \mathrm {SL}(2,\mathbb {Z})\), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\) found by Athreya and Cheung to the finite cover \(\mathrm {SL}(2,\mathbb {R})/H\) of \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erdős, Szüsz, and Turán.

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Acknowledgments

I thank my advisor Florin Boca for his guidance in this research. I thank Jayadev Athreya and Jens Marklof for constructive comments on the first draft of this paper, and I thank the referee for helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    Byron Heersink

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  1. Byron Heersink
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Correspondence to Byron Heersink.

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Communicated by S. G. Dani.

The author acknowledges support from Department of Education Grant P200A090062, “University of Illinois GAANN Mathematics Fellowship Project”.

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Heersink, B. Poincaré sections for the horocycle flow in covers of \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\) and applications to Farey fraction statistics. Monatsh Math 179, 389–420 (2016). https://doi.org/10.1007/s00605-015-0873-x

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