Deviations of ergodic sums for toral translations II. Boxes
| Autor: | Bassam Fayad, Dmitry Dolgopyat |
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| Rok vydání: | 2020 |
| Předmět: |
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Mathematics - Number Theory General Mathematics 010102 general mathematics Cauchy distribution Torus Dynamical Systems (math.DS) Space (mathematics) 01 natural sciences Combinatorics symbols.namesake Number theory Kronecker delta 0103 physical sciences FOS: Mathematics symbols Ergodic theory Number Theory (math.NT) 010307 mathematical physics Mathematics - Dynamical Systems 0101 mathematics Poisson limit theorem Mathematics |
| Zdroj: | Publications mathématiques de l'IHÉS. 132:293-352 |
| ISSN: | 1618-1913 0073-8301 |
| Popis: | We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices. Comment: 56 pages. This is a revised and expanded version of the prior submissions |
| Databáze: | OpenAIRE |
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