  We study the accuracy of the expected Euler characteristic approximation to
the distribution of the maximum of a smooth, centered, unit variance Gaussian
process f. Using a point process representation of the error, valid for
arbitrary smooth processes, we show that the error is in general exponentially
smaller than any of the terms in the approximation. We also give a lower bound
on this exponential rate of decay in terms of the maximal variance of a family
of Gaussian processes f^x, derived from the original process f.
