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 setEa,b={x∈[0,1):liminfn→∞rn(x,β)n=a,limsupn→∞rn(x,β)n=b}(0≤a≤b≤1). Furthermore, 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.