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 $b$-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 $0\le \alpha \le \mathrm{\infty }$, have full Hausdorff dimension.