  We prove sharp estimates for Fourier transforms of indicator functions of
bounded open sets in ${\mathbb R}^n$ with real analytic boundary, as well as
nontrivial lattice point discrepancy results. Both will be derived from
estimates on Fourier transforms of hypersurface measures. Relations with
maximal averages are discussed, connecting two conjectures of Iosevich and
Sawyer from [ISa1]. We also prove a theorem concerning the stability under
function perturbations of the growth rate of a real analytic function near a
zero. This result is sharp in an appropriate sense. It implies a corresponding
stability result for the local integrablity of negative powers of a real
analytic function near a zero.
