Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we employ a class of complex algebraic varieties avoiding certain regions in $\mathbb{C}^n$, which generalize hypersurfaces defined by Lee-Yang polynomials. We show that these are Delone almost periodic sets that have at most finite intersection with every discrete periodic set.

💡 Research Summary

The paper “Higher Dimensional Fourier Quasicrystals from Lee–Yang Varieties” presents a systematic method for constructing Fourier quasicrystals (FQ) with unit masses in any Euclidean dimension d. The authors generalize the one‑dimensional construction of Kurasov and Sarnak, which relied on Lee–Yang polynomials (polynomials whose zeros lie on the unit circle), to a high‑dimensional setting by introducing “strict Lee–Yang varieties”.

A strict Lee–Yang variety X⊂(ℙ¹)ⁿ is an equidimensional algebraic subvariety of dimension c=n−d satisfying three conditions: (1) invariance under coordinatewise inversion z↦1/z; (2) each point either lies on the unit torus Tⁿ or its logarithmic absolute‑value vector has at least d sign changes; (3) the torus points belong to the smooth part of X. These conditions guarantee that any point of X intersecting the torus yields only real solutions after applying a logarithmic map, mirroring the “no zeros inside/outside the unit disc” property of classical Lee–Yang polynomials.

The second ingredient is a real matrix L∈ℝⁿˣᵈ whose all d×d minors are positive. The map

  x∈ℝᵈ ↦ exp(2πiLx)∈(ℂ*)ⁿ

is then a dense immersion of ℝᵈ into the torus. The set

  Λ(X,L)= { x∈ℂᵈ | exp(2πiLx)∈X }

is shown to be real, i.e., Λ⊂ℝᵈ, whenever X is a strict Lee–Yang variety and L satisfies the positivity condition.

Main results

• Theorem 2.3 proves that Λ is a Delone set (uniformly discrete and relatively dense) and a Fourier quasicrystal. Its Fourier spectrum is explicitly described as

  Λ′ = { Lᵗk | k∈ℤⁿ, var(k)<d },

where var(k) counts sign changes after discarding zeros. Thus the spectrum is a discrete set of rational dimension ≤ n, with growth |Λ′∩B_R| = C Rⁿ + O(R^{n−1}).

• Theorem 2.5 establishes quantitative non‑periodicity. If the d×d minors of L are Q‑linearly independent and X is irreducible with X∩(ℂ*)ⁿ not a coset of an algebraic subtorus, then the Q‑span of Λ has infinite dimension. Moreover, for any set A⊂ℝᵈ whose Q‑span has dimension m, the intersection satisfies |Λ∩A| ≤ r (m+1) for a constant r depending only on n and the degree of X. Consequently, Λ cannot be contained in any finite‑rank lattice or its projection, confirming genuine high‑dimensional aperiodicity.

• Theorem 2.6 gives finer properties: Λ is Bohr almost periodic with density

  c₀ = Σ_{I∈