Approximation orders of real numbers by $β$-expansions

Lulu Fang, Min Wu, Bing Li

Published 2016-03-28Version 1

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. At last, some results related to the orbits of real numbers under $\beta$-transformation, the shrinking target type problem for $\beta$-transformations and the Diophantine-type problem for $\beta$-expansions are also given.