Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that
$$a_1 b_1 + \dots + a_n b_n = 1$$
has a solution in integers $b_1, \dots, b_n$.
I would like to get a bound saying something like:
There exists a solution with $\sum_i |b_i| < \sum_i |a_i|$
(except in the degenerate case where $a_j = 1$, $a_i = 0$ for $i \neq j$)
Presumably such things (and probably much stronger bounds) are known. Does anyone know a reference for these kinds of results?