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, multidimensional fractal sets and require only mild assumptions on the iterated function system. Moreover we provide a nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 DOI:
 10.48550/arXiv.1812.10689
 arXiv:
 arXiv:1812.10689
 Bibcode:
 2018arXiv181210689S
 Keywords:

 Mathematics  Number Theory;
 28A80;
 11H06;
 11J13
 EPrint:
 31 pages