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dc.contributor.authorElkies, Noam David
dc.contributor.authorMcMullen, Curtis T.
dc.date.accessioned2010-02-12T20:36:23Z
dc.date.issued2004
dc.identifier.citationElkies, 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.issn0012-7094en_US
dc.identifier.urihttp://nrs.harvard.edu/urn-3:HUL.InstRepos:3637161
dc.description.abstractCut 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.sponsorshipMathematicsen_US
dc.language.isoen_USen_US
dc.publisherDuke University Pressen_US
dc.relation.isversionofdoi:10.1215/S0012-7094-04-12314-0en_US
dash.licenseLAA
dc.titleGaps in \(\sqrt{n}mod 1\) and Ergodic Theoryen_US
dc.typeJournal Articleen_US
dc.description.versionAuthor's Originalen_US
dc.relation.journalDuke Mathematical Journalen_US
dash.depositing.authorMcMullen, Curtis T.
dc.date.available2010-02-12T20:36:23Z
dc.identifier.doi10.1215/S0012-7094-04-12314-0*
dash.contributor.affiliatedMcMullen, Curtis
dash.contributor.affiliatedElkies, Noam


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