Hausdorff dimension of intersections between the Jarn\'ik sets and Diophantine fractals

TL;DR

This paper investigates the fractal structure of numbers with high irrationality exponents, showing they intersect certain fractal sets with full Hausdorff dimension, revealing deep properties of Diophantine approximation.

Contribution

It establishes that the set of numbers with irrationality exponent greater than 2 intersects the limit sets of specific fractal systems with full Hausdorff dimension, highlighting new fractal properties.

Findings

The set of irrationals with irrationality exponent > 2 and bounded backward continued fraction partial quotients has Hausdorff dimension 1.

This set intersects the limit set of any parabolic iterated function system from backward continued fractions with full dimension.

No irrational number with irrationality exponent > 2 can have bounded regular continued fraction partial quotients.

Abstract

The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal structure. We show that this set intersects the limit set of any parabolic iterated function system arising from the backward continued fraction in a set of full Hausdorff dimension. As a corollary, we show that the set of irrationals whose irrationality exponents are strictly bigger than $2$ and whose backward continued fraction expansions have bounded partial quotients is of Hausdorff dimension $1$. This is a sharp contrast to the fact that there exists no irrational whose irrationality exponent is strictly greater than $2$ and whose regular continued fraction expansion has bounded partial quotients.