# Gaps in $$\sqrt{n}mod 1$$ and Ergodic Theory

 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. Full Text & Related Files: McMullen_GapsErgodoticTheory.pdf (426.9Kb; PDF) 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 Downloads of this work: