Construction of a curved Kakeya set

TL;DR

This paper constructs a measure-zero set in the plane containing segments of parabolas with varying apertures, leading to improved bounds for related maximal operators and generalizes to other curved families.

Contribution

It introduces a novel construction of a measure-zero set containing diverse parabola segments and extends this to broader families of curved lines, advancing geometric measure theory.

Findings

Constructed a measure-zero set with parabola segments of all apertures between 1 and 2.

Improved lower bounds for the $L^p$-$L^q$ maximal operator norms.

Generalized the construction to families of $C^2$ curves with curvature conditions.

Abstract

We construct a compact set in $\mathbb R^2$ of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a range of $p$, $q$. Moreover, our construction can be generalised from parabolas to a family of $C^2$ curves satisfying suitable curvature conditions.