Diophantine approximation and run-length function on β-expansions

Abstract

For any β > 1, denoted by rn(x, β) the maximal length of consecutive zeros amongst the first n digits of the β-expansion of x ∈ [0, 1]. The limit superior (respectively limit inferior) of rn(x,β) n is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level setFurthermore, we show that the extremely divergent set E0,1 which is of zero Hausdorff dimension is, however, residual. The same problems in the parameter space are also examined.

Dimension theory of Diophantine approximation related to $β$-transformations

“…The exponents ν β and νβ were introduced in [1](see also [3,Ch.7]). They are strongly related to the run-length function of β-expansions (see [19]). The aim of this paper is to study the Diophantine approximation sets in [4] when the approximation speed function n → β −nv is replaced by a general positive function.…”