Metric Diophantine approximation on the middle-third Cantor set

Abstract

Let µ ≥ 2 be a real number and let M(µ) denote the set of real numbers approximable at order at least µ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of M(µ) is equal to 2/µ. We investigate the size of the intersection of M(µ) with Ahlfors regular compact subsets of the interval [0, 1]. In particular, we propose a conjecture for the exact value of the dimension of M(µ) intersected with the middle-third Cantor set and give several results supporting this conjecture. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.

Hitting probabilities of random covering sets in tori and metric spaces

“…In the circle T 1 , the analogues of (2.5) and (2.7) were established by Li, Shieh and Xiao [40]. Bugeaud and Durand [4] also recovered these results within the context of Diophantine approximation. Li and Suomala [41] proved the analogue of (2.6) in the torus T d and showed that the assumption dim P F > d − α alone is not enough to guarantee that P(H(F )) > 0.…”

“…A problem suggested by Mahler on how well can points, say, in the middle third Cantor set K be approximated by rational points, has stemmed a number of results [5,6,18,49,57], measuring the sizes of the intersection of K with sets W(ψ) from (1.1) for different values of the approximation speed ψ. These results are in part inconclusive, and lead to conjectures on the size of the set K ∩ W(q −τ ) in [4,39]. In particular, Bugeaud and Durand [4, Conjecture 1] conjectured that…”

“…Following their reasoning, the first term in the maximum is realised for slow approximation speeds, where the intervals giving the limsup set are big, and the latter term appears when τ is large and the proportion of rationals inside the set K start to play a role. Bugeaud and Durand [4] have offered evidence that supports their conjecture, in particular, by building a model of random Diophantine approximation, where the centres of the generating intervals are independent and uniformly distributed, either in the Cantor set or in the whole circle T 1 [4, Section 2]. The random model is based on a conjecture of Broderick, Fishman and Reich [5] on the proportion of rationals in the Cantor set.…”

Uniformly de Bruijn Sequences and Symbolic Diophantine Approximation on Fractals

“…#(V (X))−ℓ δ /k . 7 In fact, the exact count for such sequences is known, but we prefer this estimate because it is simpler and yields the same upper bound on the Hausdorff dimension.…”

“…We remark that while in [12], the first-and third-named authors were able to exhibit an optimal Dirichlet function (see Definition 5.2) corresponding to Mahler's second question, it seems that finding an analogous answer to his first question is significantly harder, see e.g. [5,7,12] for detailed discussions and conjectures regarding this question.…”