Gaps in \(\sqrt{n}mod 1\) and Ergodic Theory

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Gaps in \(\sqrt{n}mod 1\) and Ergodic Theory

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Title: Gaps in \(\sqrt{n}mod 1\) and Ergodic Theory
Author: McMullen, Curtis T.; Elkies, Noam David

Note: Order does not necessarily reflect citation order of authors.

Citation: Elkies, Noam D., and Curtis T. McMullen. 2004. Gaps in √ n mod 1 and ergodic theory. Duke Mathematical Journal 123(1): 95-139. Revised 2005.
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Abstract: Cut the unit circle \(S^1 = \mathbb{R}/\mathbb{Z}\) at the points \(\{\sqrt{1}\}, \{\sqrt{2}\}, . . ., \{\sqrt{N}\}\), where \(\{x\} = x mod 1\), and let \(J_1, . . . , J_N\) denote the complementary intervals, or gaps, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths \(\mid J_i\mid/N\) are governed by an explicit piecewise real-analytic distribution \(F(t)dt\) with phase transitions at \(t=\frac{1}{2}\) and \(t=2\). The gap distribution is related to the probability \(p(t)\) that a random unimodular lattice translate \(\Lambda \subset \mathbb{R}^2\) meets a fixed triangle \(S_t\) of area \(t\); in fact \(p^"(t) = -F(t)\). The proof uses ergodic theory on the universal elliptic curve: \(E = (SL_2(\mathbb{R}) ⋉ \mathbb{R}^2) / (SL_2(\mathbb{Z}) ⋉ \mathbb{Z}^2)\)
Published Version: doi:10.1215/S0012-7094-04-12314-0
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3637161

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  • FAS Scholarly Articles [6867]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
 
 

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