Treffer: On the dynamics of the extension of a singular shift.

The author considers a two-dimensional extension \(W\) of the transformation \(T(x)=[1/x]-1/x\) as \((\xi,\eta)=W(x,y)=(T(x), 1/([1/x]-y))\) for any point \((x,y)\) in the unit square. Then the author gives a geometric algorithm setting up a one-to-one correspondence between classes of diagonally equivalent lattices and doubly infinite integer sequences, which are identified with the orbits under \(W\). He also gives a proof in the spirit of \textit{P. Lévy}'s classical work [Bull. Soc. Mat. Fr. 57, 178--194 (1929; JFM 55.0916.02)] on the regular continued fraction transformation. As an application, the author gives a metrical contribution to one-sided Diophantine approximation by proving that the set of badly approximable numbers has Lebesgue measure 0, but Hausdorff dimension 1.