Heisenberg uniqueness pairs for some algebraic curves in the plane

TL;DR

This paper investigates conditions under which measures supported on specific algebraic curves in the plane are uniquely determined by their Fourier transforms vanishing on certain sets, focusing on spirals, hyperbolas, circles, exponential curves, and parallel lines.

Contribution

It characterizes Heisenberg uniqueness pairs for various algebraic curves and reveals a new interlacing phenomenon of trigonometric polynomials for parallel lines.

Findings

Characterization of Heisenberg uniqueness pairs for spiral, hyperbola, circle, and exponential curves.

Identification of a trigonometric polynomial interlacing phenomenon for four parallel lines.

Extension of the theory to specific algebraic curves in the plane.

Abstract

A Heisenberg uniqueness pair is a pair $\left(\Gamma, \Lambda\right)$, where $\Gamma$ is a curve and $\Lambda$ is a set in $\mathbb R^2$ such that whenever a finite Borel measure $\mu$ having support on $\Gamma$ which is absolutely continuous with respect to the arc length on $\Gamma$ satisfies $\hat\mu\vert_\Lambda=0,$ then it is identically $0.$ In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.