Extravagance, irrationality and Diophantine approximation

TL;DR

This paper investigates the measure-theoretic properties of Diophantine approximation under different invariant measures for the Gauss map, revealing conditions under which almost all numbers are either Diophantine or Liouville, and constructing measures with arbitrary irrationality exponents.

Contribution

It introduces new results on the distribution of Diophantine properties under various Gauss map measures, including measures with non-integrable log partial quotients and arbitrary irrationality exponents.

Findings

Almost all numbers are Diophantine under integrable log partial quotient measures.

Almost all numbers are Liouville under non-integrable log partial quotient measures.

Existence of Gauss-invariant measures with arbitrary irrationality exponents.

Abstract

For an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotient function is integrable. We show that with respect to a ``continued fraction mixing'' measure for the Gauss map with the log of the partial quotient function non-integrable, almost all numbers are Liouville. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are applications of our study of the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type dichotomy for Diophantine approximation with respect to ergodic``weak Renyi measures'' which are ``doubling at $0$''.