On restriction of exponential sums to hypersurfaces with zero curvature

Abstract

We prove essentially sharp bounds for the $L^p$ restriction of weighted Gauss sums to monomial curves. Getting the $L^2$ upper bound combines the $TT^*$ method for matrices with the first and second derivative test for exponential sums. The matching lower bound follows via constructive interference on short blocks of integers, near the critical point of the phase function. This method is used to make the broader point that restriction to hypersurfaces is really sensitive to curvature. Our results here complement earlier results by the author and Langowski.