On the diophantine properties of λ-expansions

For and α, we consider sets of numbers x such that for infinitely many n, x is 2−αn -close to some ∑ n i=1 ω i λ i , where ω i ∈{0,1}. These sets are in Falconer’s intersection classes for Hausdorff dimension s for some s such that −(1/α)(log λ /log 2 )≤s≤1/α. We show that for almost all , the upper bound of s is optimal, but for a countable infinity of values of λ the lower bound is the best possible result.