Diophantine approximations on fractals

Details Diophantine approximations on fractals Diophantine approximations on fractals

2009-08-17

arxiv.org/abs/0908.2350v2

Mathematics

Dynamical Systems

Final version, to appear in GAFA

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Diophantine approximations on fractals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Diophantine approximations on fractals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Diophantine approximations on fractals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-19603