Diophantine approximations on fractals

TL;DR

This paper uses dynamical systems techniques to study Diophantine approximation properties on fractals, showing that typical points have rich continued fraction expansions and confirming conjectures about uniform boundedness of certain sequences.

Contribution

It introduces new methods linking dynamical properties to Diophantine approximation on fractals, proving that typical points have all finite patterns in their continued fractions and settling a conjecture about boundedness.

Findings

Almost all points on the middle third Cantor set are well approximable.

Most points on symmetric fractals in [0,1]^2 are not Dirichlet improvable.

Confirmed Boshernitzan's conjecture on boundedness of continued fractions of {nx mod 1}.

Abstract

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod1 : n is a natural number} are uniformly eventually bounded.