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.