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 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.

The Distribution of Rational Numbers on Cantor’s Middle Thirds Set

“…This conjecture was also made by Broderick, Fishman and Reich in [BFR], and futher heuristics and conjectures which imply Conjecture 1 were given by Fishman and Simmons in [FS,§ 5]. An upper bound N (T ) = O (T 2d ) was obtained by Schleischitz in [Sch,Thm. 4.1].…”

Mahler's question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory

“…For the cardinality of N n , there are some progress recently in [34,35]. But it is still far from being clear.…”

On the Arithmetic Structure of Rational Numbers in the Cantor Set

“…Namely, using the result of Korobov [6, Theorem 5] on uniformity of distribution of digits in very short segments of g-ary expansions, one can obtain finiteness results for the number of rational fractions r/q with all prime divisors of q from a fixed set S and such that |r/q − α| < exp(−(log q) 2/3+ε ) for some fixed ε > 0. Results of this type can be considered as complementary to [8,Theorem 3.4].…”

“…In particular, the choice g = 3 and D = {0, 2} corresponds to the classical Cantor set. Schleischitz [8,Corollary 4.4], among several other results on Cantor sets, has shown that for any D and any fixed set S = {p 1 , . .…”