Heisenberg uniqueness pairs for some algebraic curves and surfaces

TL;DR

This paper investigates conditions under which measures supported on algebraic curves and surfaces are uniquely determined by their Fourier transforms on specific sets, with new characterizations for parallel lines and other geometric configurations.

Contribution

It provides new characterizations of Heisenberg uniqueness pairs for algebraic curves, surfaces, and finitely many parallel lines, highlighting the role of geometric and distributional properties.

Findings

Characterization of Heisenberg uniqueness pairs for finitely many parallel lines.

Analysis of uniqueness pairs for crosses, exponential curves, and surfaces.

Dependence of determining set size on line distribution and trigonometric polynomial interlacing.

Abstract

Let $X(\Gamma)$ be the space of all finite Borel measure $\mu$ in $\mathbb R^2$ which is supported on the curve $\Gamma$ and absolutely continuous with respect to the arc length of $\Gamma$. For $\Lambda\subset\mathbb R^2,$ the pair $\left(\Gamma, \Lambda\right)$ is called a Heisenberg uniqueness pair for $X(\Gamma)$ if any $\mu\in X(\Gamma)$ satisfies $\hat\mu\vert_\Lambda=0,$ implies $\mu=0.$ We explore the Heisenberg uniqueness pairs corresponding to the cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines. We observe that the size of the determining sets $\Lambda$ for $X(\Gamma)$ depends on the number of lines and their irregular distribution that further relates to a phenomenon of interlacing of certain trigonometric polynomials.