Inferring the Kolmogorov complexity of a string from its substrings' complexity
A simple bound is that we could use a program generating $x$, a starting index $1 \leq i \leq n$ and length $m$ to get a program generating any choice of $v$, and thus $C(v)$ is bounded as
$$\forall v: k \leq C(v) \leq C(x) + \log(n) + \log(m) + c,$$
where $c$ is some constant overhead independent from $x,v$ that represents the steps necessary to take a Turing machine, a starting index and length and return the requested substring.
This in turn gives us a lower bound for $C(x)$,
$$C(x)  \geq k - \log(n) - \log(m) - c.$$