  We refer to the distance between optimal solutions of integer programs and
their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel
proved that proximity is independent of the dimension for programs in standard
form. We improve their bounds using existing and novel results on the sparsity
of integer solutions. We first bound proximity in terms of the largest absolute
value of any full-dimensional minor in the constraint matrix, and this bound is
tight up to a polynomial factor in the number of constraints. We also give an
improved bound in terms of the largest absolute entry in the constraint matrix,
after efficiently transforming the program into an equivalent one. Our results
are stated in terms of general sparsity bounds, so any new results on sparse
solutions immediately improves our work. Generalizations to mixed integer
programs are also discussed.
