Common substring with shifts in b-ary expansions

Denote by \(S_n(x,y)\) $S_n(x,y)$ 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 \(S_n(x,y)\) $S_n(x,y)$ is \(\alpha \log n\) $\alpha logn$ with \(0\le \alpha \le \infty \) $0\le \alpha \le \infty$ , have full Hausdorff dimension. Our method relies upon some estimation of the spectral radius of matrices.