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Diophantine approximations on fractals. (English) Zbl 1244.11070

From the text: “In the theory of Diophantine approximations, one wishes to understand how well vectors in \(\mathbb R^d\) can be approximated by rational vectors. The quality of approximation can be measured in various forms leading to numerous Diophantine classes of vectors such as WA (well approximable), [\(\dots\)] and so forth. Usually such a class is a null set or generic (i.e. its complement is a null set) and often one encounters the phenomena of the class being null but of full dimension. Given a closed subset \(M\subset \mathbb R^d\) supporting a natural measure [\(\dots\)], it is natural to investigate the intersection of \(M\) with the various Diophantine classes. It is natural to expect that unless there are obvious obstacles, the various Diophantine classes will intersect \(M\) in a set which will inherit the characteristics of the class, i.e. if the class is null, generic, or of full dimension in \(\mathbb R^d\), then its intersection with \(M\) would be generic, null or of full dimension in \(M\) as well.”
This paper, part of the third author’s PhD thesis at the Hebrew University of Jerusalem, deals with inheritance of genericity to certain fractals in \(\mathbb R\) or \(\mathbb R^2\), with respect to WA and other Diophantine classes. In brief, the authors prove that the above Diophantine classes remain generic or null when additional assumptions on the fractal and the measure supported on it are imposed (such as positivity of dimension and invariance under an appropriate map). To give a better information on the content of the paper, some of the results are listed below (the reviewer finds it convenient to report only those open to being stated in a simple way). In particular, the last theorem, which is of a different nature as it is an everywhere statement and whose proof involves adelic arguments, solves a conjecture by M. Boshernitzan.
Theorem 1.5. Let \(n\in \mathbb N\) and let \(\mu\) be a probability measure on the unit interval which is invariant and ergodic under multiplication by \(n\) modulo 1. Then \(\mu\)-almost any \(x\in [0,1]\) is WA.
Corollary 1.10. Almost any point in the middle third Cantor set (with respect to the natural measure) is WA and moreover its continued fraction expansion contains all patterns.
Theorem 1.11. For \(x\in [0,1]\) let \(c(x)=\limsup a_n(x)\), where \(a_n(x)\) are the coefficients in the continued fraction expansion of \(x\). Then for any irrational \(x\in [0,1]\), \(\sup_nc(\{nx\})=\infty\).

MSC:

11J83 Metric theory
11J70 Continued fractions and generalizations
11J13 Simultaneous homogeneous approximation, linear forms
37A17 Homogeneous flows
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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