Central Limit Theorems for Diophantine approximants

2018/04/17 by Michael Björklund, Björklund, Michael, Alexander Gorodnik +1 · 3 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #math.DS #math.NT #math.PR

paper · pdf · doi:10.48550/arxiv.1804.06084

50 pages

arxiv created 2018/04/17 · arxiv updated 2018/04/18

Abstract

In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior that these functions exhibit and prove a Central Limit Theorem for them. Our approach is based on a quantitative study of higher-order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of Siegel transforms.

Cited by

• Integrability of Siegel transforms and an application
• A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard
• Lévy-Khintchine Theorems: effective results and central limit theorems

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