dc.contributor.author | Elkies, Noam David | |
dc.contributor.author | McMullen, Curtis T. | |
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 | |
dc.identifier.doi | 10.1215/S0012-7094-04-12314-0 | * |
dash.contributor.affiliated | McMullen, Curtis | |
dash.contributor.affiliated | Elkies, Noam | |