期刊
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
卷 18, 期 6, 页码 1233-1272出版社
EUROPEAN MATHEMATICAL SOC
DOI: 10.4171/JEMS/612
关键词
Diophantine approximation; Hausdorff dimension; irrationality exponent; Cantor set; Mahler's problem
Let mu >= 2 be a real number and let M (mu) denote the set of real numbers approximable at order at least mu by rational numbers. More than eighty years ago, Jarnik and, independently, Besicovitch established that the Hausdorff dimension of M (mu) is equal to 2/mu. We investigate the size of the intersection of M(mu) 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 (mu) intersected with the middle-third Cantor set and give several results supporting this conjecture. We show in particular 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.
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