  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 $0\le \alpha \le \infty$, have full Hausdorff dimension.
