Ergodic Theory on Homogeneous Spaces and Metric Number Theory

2012 ◽  
pp. 302-312
Author(s):  
Dmitry Kleinbock
Keyword(s):  
Number Theory ◽  
Ergodic Theory ◽  
Homogeneous Spaces ◽  
Metric Number Theory

scholarly journals Projective metric number theory

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Anish Ghosh ◽  
Alan Haynes
Keyword(s):  
Number Theory ◽  
Projective Space ◽  
Diophantine Approximation ◽  
Higher Dimension ◽  
Probabilistic Theory ◽  
Metric Number Theory ◽  
Projective Metric

AbstractIn this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of ℚ. Using the projective metric studied in [Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 2, 211–248] we prove the analogue of Khintchine's theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin–Schaeffer conjecture.

Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory

2012 ◽  
Vol 9 (4) ◽  
pp. 2985-3059
Author(s):  
Vitaly Bergelson ◽  
Nikos Frantzikinakis ◽  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Number Theory ◽  
Ergodic Theory ◽  
Combinatorial Number Theory

Counterexamples in ergodic theory and number theory

1979 ◽  
Vol 245 (3) ◽  
pp. 185-197 ◽  
Author(s):  
Andres del Junco ◽  
Joseph Rosenblatt
Keyword(s):  
Number Theory ◽  
Ergodic Theory

scholarly journals Strong renewal theorems and Lyapunov spectra forα-Farey andα-Lüroth systems

2011 ◽  
Vol 32 (3) ◽  
pp. 989-1017 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
SARA MUNDAY ◽  
BERND O. STRATMANN
Keyword(s):  
Number Theory ◽  
Ergodic Theory ◽  
Level Sets ◽  
Gauss Map ◽  
Unit Interval ◽  
Elementary Number Theory ◽  
The Core ◽  
Lyapunov Spectra ◽  
Infinite Ergodic Theory ◽  
Farey Map

AbstractIn this paper, we introduce and study theα-Farey map and its associated jump transformation, theα-Lüroth map, for an arbitrary countable partitionαof the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have calledα-sum-level sets for theα-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of theα-Farey map and theα-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partitionα.

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