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On intrinsic and extrinsic rational approximation to Cantor sets

Part of: Diophantine approximation, transcendental number theory Geometry of numbers Classical measure theory

Published online by Cambridge University Press:  10 February 2020

JOHANNES SCHLEISCHITZ
JOHANNES SCHLEISCHITZ*
Affiliation:
Middle East Technical University, Northern Cyprus Campus, Kalkanli, Güzelyurt, Turkey email [email protected]
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Abstract

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. 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 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; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

Keywords

Cantor setsintrinsic/extrinsic Diophantine approximationcontinued fractions

MSC classification

Primary: 28A80: Fractals 11J13: Simultaneous homogeneous approximation, linear forms
Secondary: 11H06: Lattices and convex bodies
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1560 - 1589
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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