Diophantine approximation and self-conformal measures
The author considers metric Diophantine approximation using some novel measures on sets of a fractal character. A point \({\mathbf x}= (x_1,\dots, x_n)\in\mathbb{R}^n\) is said to be very well approximable (VWA) if for some \(\delta> 0\), there are infinitely many \({\mathbf p}= (p_1,\dots, p_n)\in\mathbb{Z}^n\) and \(q\in\mathbb{N}\) such that \[ |q{\mathbf x}-{\mathbf p}|_\infty= \max\{|qx_i- p_i|: i= 1,\dots, n\}\leq q^{-{1\over n}-\delta} \] (the notation \(\| q{\mathbf x}-{\mathbf p}\|\) used by the author presumably means the \(L^2\) norm but for the sets involved, it is immaterial whether this or the more usual \(L^\infty\) norm is used). The point \({\mathbf x}\) is multiplicatively VWA if the weaker product inequality \[ \prod^n_{i=1} |qx_i- p_i|\leq q^{-1-\delta} \] holds infinitely often. Khintchine's theorem on metric Diophantine approximation in \(\mathbb{R}^n\) implies that almost all points are not VWA (i.e., the set of VWA points is a set of Lebesgue measure zero) or that \(\mathbb{R}^n\) is extremal and \textit{P. X. Gallagher} [J. Lond. Math. Soc. 37, 387--390 (1962; Zbl 0124.02902)] proved that almost no points are multiplicatively VWA or that \(\mathbb{R}^n\) is strongly extremal. The Hausdorff dimension of the set of VWA points is \(n(n+ 1)/(n+ 1 +\delta)\) [\textit{V. Jarnik}, Math. Z. 33, 505--543(1931; Zbl 0001.32403)] and not \(n\) as stated; that of the multiplicative VWA set is \(n- 1+ 2/(2+ \delta)\) [\textit{J., D. Bovey} and \textit{M. M. Dodson}, Bull. Lond. Math. Soc. 10, 213--218 (1978; Zbl 0384.10013)]. \textit{D. Y. Kleinbock}, \textit{E. Lindenstrauss} and \textit{B. Weiss} studied an extension of this notion to measures [On fractal measures and Diophantine approximation, Sel. Math. (N.S.), 10, No. 4, 479--523 (2004; Zbl 1130.11039)] a Borel measure \(\mu\) on \(\mathbb{R}^n\) is said to be (strongly) extremal if \(\mu\)-almost all points are not (multiplicatively) VWA; friendly measures are introduced and shown to be strongly extremal. Absolutely friendly measures are identified (the name is from [\textit{A. Pollington} and \textit{S. L. Velani}, Metric Diophantine approximation and ``absolutely'' friendly measures, to appear in Selecta Math.]). In addition to the nice properties of friendly measures, absolutely friendly meaures enjoy globally well controlled intersections of `balls' with `thickened' hyperplanes. Kleinbock et al. showed that the Hausdorff measure on the limit set of a finite iterated function system is absolutely friendly and hence strongly extremal; and they conjectured that this result also holds for conformal systems. By proving a more general result concerning Gibbs states of Hölder families of functions, the present author establishes their conjecture for finite conformal irreducible iterated function systems \(S\) in \(\mathbb{R}^d\), where \(d\geq 2\) (to avoid having to assume the bounded distortion lemma). He also deduces the strong extremality of measures invariant with respect to the ordered pair \((S, P)\), where \(P\) is a probability vector.
scientific article; zbMATH DE number 2153780