The PAC learning of rectangles has been studied because they have
been found experimentally to yield excellent hypotheses for several
applied learning problems. Also, pseudorandom sets for rectangles
have been actively studied recently because (i) they are a subproblem
common to the derandomization of depth-2 (DNF) circuits and
derandomizing Randomized Logspace, and (ii) they approximate the
distribution of independent multivalued random variables. We present
improved upper bounds for a class of such problems of "approximating"
high-dimensional rectangles that arise in PAC learning and
pseudorandomness.