42 research outputs found

    On a Problem in Diophantine Approximation

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    We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.Comment: 16 page

    Separating Solution of a Quadratic Recurrent Equation

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    In this paper we consider the recurrent equation Λp+1=1pq=1pf(qp+1)ΛqΛp+1q\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q} for p1p\ge 1 with fC[0,1]f\in C[0,1] and Λ1=y>0\Lambda_1=y>0 given. We give conditions on ff that guarantee the existence of y(0)y^{(0)} such that the sequence Λp\Lambda_p with Λ1=y(0)\Lambda_1=y^{(0)} tends to a finite positive limit as pp\to \infty.Comment: 13 pages, 6 figures, submitted to J. Stat. Phy

    Ergodic Properties of Square-Free Numbers

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    We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.Comment: 37 page, 1 figur

    Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems

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    In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in pp-variation and c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the Conference in Honor of the 75th Birthday of S.R.S. Varadha