On fractal measures and Diophantine approximation
The paper under review is concerned with an extension of Mahler's problem in metric number theory, originally resolved by \textit{V. G. Sprindzuk} [''Mahler's problem in metric number theory,'' Nauka i Teknika, Minsk (1967; Zbl 0168.29504)]. Mahler conjectured that almost no points with respect to the natural measure on the curve \((x, x^2, \dots, x^n)\) should be no better approximable by rationals with the same denominator than is the case for generic points in \(\mathbb R^n\). Here the approximation is measured in the sup-norm. A subset \(E\) of \(\mathbb R^n\) supporting a natural measure is said to be \textit{extremal} if it satisfies this property. If we instead consider multiplicative approximation, where the distances of the individual coordinates are multiplied, we arrive at the notion of \textit{strong extremality}, which implies extremality. In a far-reaching extension of Mahler's conjecture, \textit{D. Y. Kleinbock} and \textit{G. A. Margulis} [Ann. Math. (2) 148, No.1, 339--360 (1998; Zbl 0922.11061)] showed that manifolds satisfying a certain non-degeneracy condition are strongly extremal. The present paper extends and resolves Mahler's conjecture further. We say that a measure is (strongly) extremal if the support of the measure satisfies the (strong) extremality property. A measure \(\mu\) on \(\mathbb R^n\) is said to be \textit{friendly} if it is doubling (or Federer) almost everywhere, if the \(\mu\)-measure of any affine hyperplane is zero, and if the measure satisfies a certain technical decay condition. The main result of the paper under review is that friendly measures are strongly extremal. The class of friendly measures include the volume measure on non-degenerate manifolds, and so the result of \textit{Kleinbock} and \textit{Margulis} [loc.cit.] is contained in the main result. In fact, this is part of a larger class of push-forwards of the so-called absolutely friendly measures. Other examples are also given. The Hausdorff \(s\)-measure restricted to an \(s\)-dimensional attractor of an iterated function system of affine contractions satisfying the open set condition is friendly. The same holds for direct products of friendly measures. The main theorem is proved by extending the method of \textit{Kleinbock} and \textit{Margulis} [loc.cit.], re-interpreting the strong extremality property as a certain quantitative non-divergence property for a certain flow in the homogeneous space \( \text{SL}(n,\mathbb R)/\text{SL}(n, \mathbb Z)\). It is then shown that the above measures are all friendly. The paper is concluded with a section on related problems, results and conjectures.
scientific article; zbMATH DE number 2168146