# Hausdorff dimension of a class of three-interval exchange maps

## TL;DR

This paper improves the Diophantine conditions for a class of three-interval exchange maps related to Sarnak's conjecture and analyzes the Hausdorff dimension of the parameter set, showing it has positive but not full measure.

## Contribution

It refines Bourgain's conditions and estimates the Hausdorff dimension of the parameter set, revealing it has positive measure but zero Lebesgue measure.

## Key findings

- The parameter set has positive Hausdorff dimension.
- The parameter set has zero Lebesgue measure.
- The Diophantine condition is slightly improved.

## Abstract

In \cite{B} Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of Three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show, that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.