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

Abstract

In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set K, i.e. approximating the points in K by rational numbers inside K, a question posed by K. Mahler. By using another height function of a rational number in K, i.e. the denominator obtained from its periodic 3-adic expansion, a complete metric theory for this variant intrinsic Diophantine approximation is presented which yields the divergence theory of Mahler's original question.

Approximating elements of the middle third Cantor set with dyadic rationals

“…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.…”

Intrinsic Diophantine Approximation for overlapping iterated function systems

“…Remark 1.2. Statement 3 of Theorem 1.1 was proved for the IFS {φ 1 (x) = x 3 , φ 2 (x) = x+2 3 } by Tan, Wang, and Wu in [37]. Note that this IFS has the middle third Cantor set as its self-similar set.…”

“…The proof given in [37] can be generalised to prove an analogue of Theorem 2.2 for any equicontractive IFS acting on R satisfying the strong separation condition. Our main result in this direction is the following statement.…”

“…The divergence condition they needed for a full measure statement was not optimal. This issue was addressed in a recent paper by Tan, Wang, and Wu [37] who established a complete analogue of Khintchine's theorem for the set…”

“…A tremendous amount of work has been done on these two topics when X is taken to be such a set. For further details we refer the reader to the papers [7,8,9,10,12,13,15,16,18,17,25,26,34,35,37,39,40] and the references therein. In this paper we study intrinsic Diophantine Approximation when the set X is a self-similar set.…”