Restriction of exponential sums to hypersurfaces

TL;DR

This paper establishes sharp moment inequalities for exponential sums restricted to hypersurfaces, matching Fourier decay properties, with some results achieving lossless estimates.

Contribution

It introduces new sharp moment inequalities for exponential sums on hypersurfaces, improving upon previous bounds and removing losses in certain cases.

Findings

Sharp moment inequalities for exponential sums on hypersurfaces

Matching Fourier decay properties of curved hypersurfaces

Some estimates achieved without loss factors

Abstract

We prove moment inequalities for exponential sums with respect to singular measures, whose Fourier decay matches those of curved hypersurfaces. Our emphasis will be on proving estimates that are sharp with respect to the scale parameter $N$, apart from $N^\epsilon$ losses. In a few instances, we manage to remove these losses.