The Hausdorf dimension of certain sets in a class of α-Lroth expansions

In this paper,we consider sets of points with some restricts on the digits of theirα-Lroth expansions.More precisely,for any countable partitionα={An,n∈N}of the unit interval I,we completely determine the Hausdorf dimensions of the sets F(α,φ)=x=l1(x),l2(x),...]α∈I:ln(x)φ(n),n 1,whereφis an arbitrary positive function defined on N satisfyingφ(n)→∞as n→∞.