An effective estimate for the Lebesgue measure of preimages of iterates of the Farey map

TL;DR

This paper provides an effective estimate for the Lebesgue measure of preimages of Farey map iterates, improving previous asymptotic results by employing transfer operator techniques and an effective Tauberian theorem.

Contribution

It offers a new, effective version of the measure estimate for Farey map preimages using basic transfer operator properties and Freud's effective Tauberian theorem.

Findings

Derived explicit bounds for Lebesgue measure of preimages

Extended asymptotic results to effective estimates

Applied transfer operator and Tauberian techniques

Abstract

Using techniques from infinite ergodic theory, Kessebohmer and Stratmann determined the asymptotic behavior of the Lebesgue measure of sets of the form $F^{-n}[\alpha,\beta]$, where $[\alpha,\beta]\subseteq(0,1]$ and $F$ is the Farey map. In this paper, we provide an effective version of this result, employing mostly basic properties of the transfer operator of the Farey map and an application of Freud's effective version of Karamata's Tauberian theorem.