# Small-Support Uncertainty Principles on $\mathbb{Z}/p$ over Finite Fields

## TL;DR

This paper proves an uncertainty principle for functions over finite fields with small support, showing that their Fourier transforms must have nearly full support, using Szemeredi's theorem and Gowers' improvements.

## Contribution

It establishes a new uncertainty principle for functions on finite fields with constant support, extending previous results with advanced combinatorial methods.

## Key findings

- Functions with constant support have Fourier transforms covering almost entire domain.
- The proof uses Szemeredi's theorem and Gowers' results to strengthen the uncertainty bounds.
- Supports of functions and their Fourier transforms are tightly linked in finite field settings.

## Abstract

We establish an uncertainty principle for functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ with constant support (where $p \mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ for which $|\text{supp}\; {f}| = S$ must satisfy $|\text{supp}\; \hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredi's theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.