A dichotomy law for the diophantine properties in \(\beta\)-dynamical systems (Q2827921)

For a real number \(\beta > 1\), the \(\beta\)-expansion map, \(T_\beta : [0,1] \rightarrow [0,1]\), is given by \(T_\beta x = \{\beta x\}\), where \(\{ \cdot \}\) denotes the fractional part. The authors provide zero-infinity laws for the Hausdorff measure of the set NEWLINE\[NEWLINE W_y(T_\beta, \Psi) =\{ x \in [0,1] : | T_\beta^n x - y | < \Psi(n) \text{ for infinitely many } n \in\mathbb N\},NEWLINE\]NEWLINE where \(y \in [0,1]\) is fixed and \(\Psi :\mathbb N\rightarrow\mathbb R_+\) is a function tending to zero as the argument tends to infinity. Additionally, they obtain a zero-infinity law for the `doubly metric' variant, NEWLINE\[NEWLINE W(T_\beta, \Psi) =\{ (x,y) \in [0,1]^2 : | T_\beta^n x - y | < \Psi(n) \text{ for infinitely many } n \in \mathbb N\}, NEWLINE\]NEWLINE The measure is zero or infinity depending on the convergence or divergence of certain series.NEWLINENEWLINEThe results extend those of \textit{W. Philipp} [Pac. J. Math. 20, 109--127 (1967; Zbl 0144.04201)], who obtained a zero-one law for the Lebesgue measure of a variant of the set \(W_y(T_\beta, \Psi)\). The proofs depend heavily on this result together with the mass transference and slicing techniques developed by \textit{V. Beresnevich} and \textit{S. Velani} [Ann. Math. (2) 164, No. 3, 971--992 (2006; Zbl 1148.11033); Int. Math. Res. Not. 2006, No. 19, Article ID 48794, 24 p. (2006; Zbl 1111.11037)].