  A classic result of Cook et al. (1986) bounds the distances between optimal
solutions of mixed-integer linear programs and optimal solutions of the
corresponding linear relaxations. Their bound is given in terms of the number
of variables and a parameter $ \Delta $, which quantifies sub-determinants of
the underlying linear inequalities. We show that this distance can be bounded
in terms of $ \Delta $ and the number of integer variables rather than the
total number of variables. To this end, we make use of a result by Olson (1969)
in additive combinatorics and demonstrate how it implies feasibility of certain
mixed-integer linear programs. We conjecture that our bound can be improved to
a function that only depends on $ \Delta $, in general.
