Test if there exists an integer k to add to one sequence to make it a subsequence of another sequence
Here is a heuristic that won't always work, but should work with high probability if the integers in the arrays are chosen randomly from a large enough space.

Initialize a hashtable of counts $C$ to all zeros.  Then, repeat $t$ times: randomly pick $i,j$, compute $b_j-a_i$, and increment $C[b_j-a_i]$.  Finally, sort $C$ by counts, from largest count to smallest; then for each of the largest few values of $C[k']$, try $k'$ as your guess at $k$, and verify each guess.

Notice that in each iteration, the probability of incrementing $C[k]$ is at least $1/m$; whereas if $l\ne k$, we expect $C[l]$ to be incremented much more rarely (assuming the integers in the arrays are random and large enough).  Thus, after $t$ iterations, we expect $\mathbb{E}[C[k]] \ge t/m$ but $\mathbb{E}[C[l]] \ll t/m$.  So, once $t$ is large enough, $C[k]$ should be larger than every other entry in $C$.

How large does $t$ need to be?  I expect that $t = O(m \log m)$ should suffice, based on a central limit theorem approximation for the counts $C[l]$, assuming we are willing to accept a small probability of error (which can be driven exponentially small).  The constant factor hidden by the big-O notation might be non-trivial.

This is only a heuristic, and there will certainly be inputs where it fails.