Diophantine properties of measures invariant with respect to the Gauss map

TL;DR

This paper investigates the Diophantine properties of measures invariant under the Gauss map, establishing conditions under which such measures are extremal and providing counterexamples related to measure regularity and approximability.

Contribution

It proves that invariant measures with finite Lyapunov exponent are extremal and constructs examples with infinite Lyapunov exponent that are not extremal, answering a key open question.

Findings

Measures with finite Lyapunov exponent are extremal.

Existence of measures with infinite Lyapunov exponent that are not extremal.

Counterexamples of Ahlfors regular measures not satisfying a 0-1 law.

Abstract

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we answer in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss, by constructing a family of measures on the real line which are Ahlfors regular and yet do not satisfy a 0-1 law for approximability.