On intrinsic and extrinsic rational approximation to Cantor sets

Abstract

We establish various new results on a problem proposed by K. Mahler in 1984 concerning rational approximation to fractal sets by rational numbers inside and outside the set in question, respectively. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We want to highlight two of them: Firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets. Secondly we find properties of the denominator structure of rational points in ''missing digit'' Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

31 pages

References in corpus (2)

• An analogue of Khintchine's theorem for self-conformal sets
• Some Existence Results on Cantor Sets

Cited by in corpus (3)

• Rational Points in Translations of The Cantor Set
• Zero-full law for well approximable sets in missing digit sets
• On a class of self-similar sets which contain finitely many common points