On restriction of exponential sums to hypersurfaces with zero curvature

TL;DR

This paper establishes sharp bounds for the restriction of weighted Gauss sums to monomial curves, highlighting the critical role of curvature in restriction phenomena and employing advanced exponential sum techniques.

Contribution

It provides the first sharp bounds for $L^p$ restriction of Gauss sums to monomial curves with zero curvature, combining matrix analysis and exponential sum estimates.

Findings

Sharp $L^p$ bounds for Gauss sums on monomial curves

Restriction sensitivity to hypersurface curvature demonstrated

Methodology combines $TT^*$ and derivative tests

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.