Suppose I have integers $ a_1, …, a_n$ which are coprime, meaning that

$ $ a_1 b_1 + … + a_n b_n = 1$ $

has a solution in integers $ b_1, …, 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?