  Given an $n\times n$ matrix with integer entries in the range $[-h,h]$, how
close can two of its distinct eigenvalues be?
  The best previously known examples have a minimum gap of $h^{-O(n)}$. Here we
give an explicit construction of matrices with entries in $[0,h]$ with two
eigenvalues separated by at most $h^{-n^2/16+o(n^2)}$. Up to a constant in the
exponent, this agrees with the known lower bound of
$\Omega((2\sqrt{n})^{-n^2}h^{-n^2})$ \cite{mahler1964inequality}. Bounds on the
minimum gap are relevant to the worst case analysis of algorithms for
diagonalization and computing canonical forms of integer matrices.
  In addition to our explicit construction, we show there are many matrices
with a slightly larger gap of roughly $h^{-n^2/32}$. We also construct 0-1
matrices which have two eigenvalues separated by at most $2^{-n^2/64+o(n^2)}$.
