Common substring with shifts in b-ary expansions

Denote by S-n(x,y) the length of the longest common substring of x and y with shifts in their first n digits of the b-ary expansions. We show that the sets of pairs (x, y), for which the growth rate of S-n(x,y) is alpha log n with 0 <= alpha <= infinity, have full Hausdorff dimension. Our method relies upon some estimation of the spectral radius of matrices.