On short sums of trace functions

TL;DR

This paper develops new bounds for sums of oscillating functions over intervals in cyclic groups near the square root size, and shows these bounds are stable under Fourier transform, with applications to trace functions over finite fields.

Contribution

It provides non-trivial estimates for sums of bounded functions with bounded Fourier transforms on intervals slightly larger than the square root, and demonstrates stability under Fourier transform.

Findings

Established bounds for sums on intervals just above the square root threshold.

Proved stability of these bounds under the discrete Fourier transform.

Applied results to trace functions over finite fields.

Abstract

We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Poly\'a-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform, and we give applications related to trace functions over finite fields.