Uncertainty in finite planes

TL;DR

This paper proves new uncertainty inequalities for functions on finite affine planes, sharpening existing bounds by considering support structures, and shows that supports of a function and its Fourier transform cannot both be too small.

Contribution

It improves Meshulam's bound for finite affine planes by incorporating support structure, providing sharper uncertainty inequalities.

Findings

Supports of functions and their Fourier transforms cannot both be small.

Established a lower bound of approximately 3p(p-2) for the product of support sizes.

Classified exceptions where the bound does not hold.

Abstract

We establish a number of uncertainty inequalities for the additive group of a finite affine plane, showing that for $p$ prime, a nonzero function $f\colon\mathbb F_p^2\to\mathbb C$ and its Fourier transform $\hat f\colon\widehat{\mathbb F_p^2}\to\mathbb C$ cannot have small supports simultaneously. The "baseline" of our investigation is the well-known Meshulam's bound, which we sharpen, for the particular groups under consideration, taking into account not only the sizes of the support sets $\mathrm{supp}\,f$ and $\mathrm{supp}\,\hat f$, but also their structure. Our results imply in particular that, with some explicitly classified exceptions, one has $|\mathrm{supp}\,f||\mathrm{supp}\,\hat f|\ge3p(p-2)$; in comparison, the classical uncertainty inequality gives $|\mathrm{supp}\,f||\mathrm{supp}\,\hat f|\ge p^2$.