Let be a real number. Let denote the -transformation on . A cylinder of order is a set of real numbers in having the same first digits in their -expansion. A cylinder is called full if it has maximal length, i.e., if its length is equal to . In this paper, we show that full cylinders are well distributed in in a suitable sense. As an application to the metrical theory of -expansions, we determine the Hausdorff dimension of the set
where is a sequence of real numbers in , the function is continuous, and denotes the ergodic sum .
Cite this article
Yann Bugeaud, Bao-Wei Wang, Distribution of full cylinders and the Diophantine properties of the orbits in -expansions. J. Fractal Geom. 1 (2014), no. 2, pp. 221–241DOI 10.4171/JFG/6