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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dc.contributor.author McMullen, Curtis T.
dc.contributor.author Elkies, Noam David
dc.date.accessioned 2010-02-12T20:36:23Z
dc.date.issued 2004
dc.identifier.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. en_US
dc.identifier.issn 0012-7094 en_US
dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:3637161
dc.description.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: \(E = (SL_2(\mathbb{R}) ⋉ \mathbb{R}^2) / (SL_2(\mathbb{Z}) ⋉ \mathbb{Z}^2)\) en_US
dc.description.sponsorship Mathematics en_US
dc.language.iso en_US en_US
dc.publisher Duke University Press en_US
dc.relation.isversionof doi:10.1215/S0012-7094-04-12314-0 en_US
dash.license LAA
dc.title Gaps in \(\sqrt{n}mod 1\) and Ergodic Theory en_US
dc.type Journal Article en_US
dc.description.version Author's Original en_US
dc.relation.journal Duke Mathematical Journal en_US
dash.depositing.author McMullen, Curtis T.
dc.date.available 2010-02-12T20:36:23Z

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  • FAS Scholarly Articles [6463]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University

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