Ubiquity and a general logarithm law for geodesics

Victor Beresnevich; Sanju Velani; Y. Bugeaud; F. Dal'Bo; C. Drutu

Ubiquity and a general logarithm law for geodesics

Victor Beresnevich, Sanju Velani, Y. Bugeaud (Editor), F. Dal'Bo (Editor), C. Drutu (Editor)

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of 'local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in tememoirs. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in tememoirs.

Original language	English
Title of host publication	Dynamical Systems and Diophantine Approximation. Seminaires et Congres
Pages	21-36
Number of pages	16
Volume	19
Publication status	Published - 2009

Publication series

Name	Seminaires et Congres

Keywords

Number Theory

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title = "Ubiquity and a general logarithm law for geodesics",

abstract = "There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of 'local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in tememoirs. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in tememoirs.",

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TY - CHAP

T1 - Ubiquity and a general logarithm law for geodesics

AU - Beresnevich, Victor

AU - Velani, Sanju

A2 - Bugeaud, Y.

A2 - Dal'Bo, F.

A2 - Drutu, C.

PY - 2009

Y1 - 2009

N2 - There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of 'local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in tememoirs. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in tememoirs.

AB - There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of 'local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in tememoirs. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in tememoirs.

KW - Number Theory

M3 - Chapter

SN - 978-2-85629-303-4

VL - 19

T3 - Seminaires et Congres

SP - 21

EP - 36

BT - Dynamical Systems and Diophantine Approximation. Seminaires et Congres

ER -