Heisenberg uniqueness pairs for some algebraic curves in the plane

Abstract

Abstract. A Heisenberg uniqueness pair is a pair (Γ, Λ), where Γ is a curve and Λ is a set in R 2 such that whenever a finite Borel measure µ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfiesμ| Λ = 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.

Heisenberg uniqueness pairs for some algebraic curves and surfaces

“…In a recent article [22], the second author has shown that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely lay on the level surface of any homogeneous harmonic polynomial on R n . Thereafter, a sense of evidence emerged that the exceptional sets for the HUPs corresponding to the sphere are eventually contained in the zero sets of the spherical harmonics and the Bessel functions, though we yet resolve it (see [6,22]).…”

“…Subsequently, Babot [2] has given a characterization of the Heisenberg uniqueness pairs corresponding to a certain system of three parallel lines. Thereafter, the authors in [6] have given some necessary and sufficient conditions for the Heisenberg uniqueness pairs corresponding to a system of four parallel lines. However, an exact analogue for the finitely many parallel lines is still unsolved.…”

“…Afterward, the authors have given the following characterization for the Heisenberg uniqueness pairs corresponding to a certain system of four parallel lines (see [6]). Let Γ denotes a system of four parallel straight lines in R 2 which can be represented by Γ = R × {α, β, γ, δ}, where α < β < γ < δ with (γ − α)/(β − α) = 2 and (δ − α)/(β − α) ∈ N. By the invariance properties of HUP, one can assume that Γ = R × {0, 1, 2, p} for some p ≥ 3.…”

“…Theorem 1.6. [6] Let Γ = R × {0, 1, 2, p} and Λ ⊂ R 2 be a closed set which is 2-periodic with respect to the second variable. Suppose Π(Λ) is dense in R. If (Γ, Λ) is a Heisenberg uniqueness pair, then the set…”

Heisenberg uniqueness pairs corresponding to a finite number of parallel lines

“…For example, the case of circle and parabola were studied by Nir Lev [6] and Per Sjölin [8] respectively. Please see [2], [4], [7], [9] and [10] for more results in this direction.…”