Dyadic Approximation in the Middle-Third Cantor Set

Abstract

In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley, Salp, and Velani (2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured.

Approximating elements of the middle third Cantor set with dyadic rationals

“…Conjecture 1.2 predicts that a µ typical point will exhibit the same behaviour as a Lebesgue typical point. In the special case where ψ is monotonic and x takes the constant value zero, Conjecture 1.2 is attributed to Velani in [2]. We expect this conjecture to be true in this more general framework so formulate it this way.…”

“…These two questions have generated a substantial amount of research (see [2,3,4,5,7,10,11,12,21,23,24,29,31,32,33,35] and the references therein). We do not attempt to give an exhaustive overview of research in this area.…”

The Divergence Borel-Cantelli Lemma revisited

“…The divergence part is also known as the second Borel-Cantelli Lemma and it naturally shows up (in some form) in the proof of the notorious Duffin-Schaeffer Conjecture [17] recently given by Koukoulopoulos & Maynard [25] and its higher dimensional generalisation proved two decades earlier by Pollington & Vaughan [30]. Indeed, the divergence Borel-Cantelli Lemma is very much at the heart of numerous other recent advances on topical problems in metric number theory, such as those in the theory of multiplicative and inhomogeneous Diophantine approximation and Diophantine approximation on manifolds and more generally on fractals, see for example [1,5,12,13,14,15,23,32,33,36]. In a nutshell, our goal it is to revisit the Borel-Cantelli Lemma and to establish both sufficient and necessary conditions that guarantee either positive or full measure.…”

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

“…For example, in [18], Fishman & Simmons considered the extrinsic Diophantine approximation, i.e. Mahler's second question; in [27], Kristensen considered the approximation of points in K by algebraic numbers; in [1], Allen, Chow & Yu considered the approximating the points in K by dyadic rational numbers. There are also two excellent works without restricting the rationals inside the Cantor set: in [21] Khalil & Luethi established the Khintchine's theorem on some fractals, for example, generated by similarities and with sufficiently large Hausdorff dimension; in [19] Han established the Khintchine's theorem for some fractals with large l 1 -dimension, for example middle b-adic Cantor set with b large.…”

“…As a consequence, by writing W (1) (ψ) = x ∈ K : |x−p/q| < ψ(q), i.m. p/q reduced, p/q ∈ K, P(p/q) ≤ log 2 log q we have the following, Corollary 6.5.…”