Intrinsic approximation for fractals defined by rational iterated function systems - Mahler's research suggestion
Number Theory
2017-05-17 v3
Abstract
Following K. Mahler's suggestion for further research on intrinsic approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterated function systems. We further investigate the behavior of these approximation functions under random translations. We connect the information regarding the distribution of rationals on the limit set encoded in the system to the distribution of rationals in reduced form by proving a Khinchin type theorem. Finally, using a result of S. Ramanujan, we prove a theorem motivating a conjecture regarding the distribution of rationals in reduced form on the Cantor ternary set.
Keywords
Cite
@article{arxiv.1208.2089,
title = {Intrinsic approximation for fractals defined by rational iterated function systems - Mahler's research suggestion},
author = {Lior Fishman and David Simmons},
journal= {arXiv preprint arXiv:1208.2089},
year = {2017}
}