On a problem of K. Mahler: Diophantine approximation and Cantor sets. (English) Zbl 1115.11040

Consider the Cantor middle third set \(K\), i.e., the set of numbers in \([0,1]\) which can be expressed in base \(3\) without use of the digit \(1\). In the paper under review, the authors consider the approximation of numbers in \(K\) by rational numbers, whose denominator is a power of \(3\). For a real, positive function \(\psi\), the set \(W_{\mathcal A}(\psi)\) consists of the numbers \(x \in [0,1]\) for which \[ | x-p/q| < \psi(q), \] for infinitely many \(p,q \in \mathbb{Z}\) with \(q = 3^n\). A complete metrical theory is developed for the sets \(W_{\mathcal A}(\psi) \cap K\), in the sense that the Hausdorff \(f\)-measure is zero or infinity according to the convergence or divergence of a certain series. It is interesting to note that no monotonicity assumptions are needed on the error function \(\psi\). As a corollary to the metrical result, the authors answer a question attributed to Mahler, namely they prove the existence of very well approximable numbers other than Liouville numbers in the Cantor set. The metrical theory is easily extended to other ‘missing digit’ sets.
Recall that the irrationality exponent \(\mu(x)\) of a number \(x\) is defined to be the supremum of all \(\mu > 0\) for which the inequality \[ | x-p/q| <q^{-\mu}, \] has infinitely many solutions \(p/q \in \mathbb{Q}\). For any \(x\), \(\mu(x) \geq 2\). In addition to the metrical theory, the authors construct explicit examples of numbers in \(K\) with any prescribed irrationality exponent greater than \((3+\sqrt{5})/2\).
Finally, the authors discuss the case when the denominators of the approximating rationals are not required to be powers of \(3\). Some conjectures are given in this direction, and the question of the existence of algebraic irrationals inside \(K\) (also attributed to Mahler) is also discussed. Both of these latter problems remain open.


11J83 Metric theory
11J82 Measures of irrationality and of transcendence
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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