The Fourier transform of planar convex bodies and discrepancy over intervals of rotations

TL;DR

This paper investigates the Fourier transform of planar convex bodies averaged over affine transformations and analyzes their discrepancy over rotation intervals, providing bounds and asymptotic behavior insights.

Contribution

It introduces new bounds on Fourier transforms of convex bodies and resolves an open question on affine quadratic discrepancy over rotation intervals.

Findings

Established bounds on Fourier transforms based on geometric properties.

Provided sharp asymptotic results for affine quadratic discrepancy.

Answered an open question by Bilyk and Mastrianni regarding rotation averages.

Abstract

This work studies the Fourier transform of the characteristic function of planar convex bodies averaged over affine transformations. We establish lower and upper bounds on the latter quantities in terms of the geometric properties of the bodies considered. The second matter of study is the affine quadratic discrepancy of planar convex bodies, and we present sharp results on its asymptotic behaviour. In particular, we address averages over intervals of rotations, answering an open question of Bilyk and Mastrianni.