Complex Hadamard matrices and the Spectral Set Conjecture

Abstract

By analyzing the connection between complex Hadamard matrices and spectral sets we prove the direction ``spectral -> tile'' of the Sectral Set Conjecture for all sets A of size at most 5 in any finite Abelian group. This result is then extended to the infinite grid ${\mathbb{Z}}^d$ for any dimension d, and finally to Euclidean space. It was pointed out recently by Tao that the corresponding statement fails for |A|=6 in the group ${\mathbb{Z}}_3^5$, and this observation quickly led to the failure of the Spectral Set Conjecture in ${\mathbb{R}}^5$ (Tao), and subsequently in ${\mathbb{R}}^4$ (Matolcsi). In the second part of this note we reduce this dimension further, showing that the direction ``spectral -> tile'' of the Spectral Set Conjecture is false already in dimension 3. In a computational search for counterexamples in lower dimension (one and two) one needs, at the very least, to be able to decide efficiently if a set is a tile (in, say, a cyclic group) and if it is spectral. Such efficient procedures are lacking however and we make a few comments for the computational complexity of some related problems.

Cited by in corpus (39)

• Spectral property of self-affine measures on ${\mathbb{R}}^n$
• Fuglede's conjecture fails in 4 dimensions over odd prime fields
• Equi-distributed property and spectral set conjecture on ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$
• Hadamard triples generate self-affine spectral measures
• Fuglede's conjecture holds for cyclic groups of order $pqrs$
• On the structure of spectral and tiling subsets of cyclic groups
• Spectrality of random convolutions generated by finitely many Hadamard triples
• Distances sets that are a shift of the integers and Fourier basis for planar convex sets
• A new method to construct families of complex Hadamard matrices in even dimensions
• Self-affine spectral measures and frame spectral measures on ${\mathbb{R}}^d$
• Fuglede's conjecture holds in ${\mathbb{Q}}_p$
• Tilings by translation
• Spectrality of Infinite Convolutions and Random Convolutions
• Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions
• Addendum to "Fuglede's conjecture fails in 4 dimensions over odd prime fields"
• Packing near the tiling density and exponential bases for product domains
• The spectrality of self-affine measure under the similarity transformation of $G{L}_n(p)$
• Compact Open Spectral Sets In ${\mathbb{Q}}_p$
• A note on the existence of BH(19,6) matrices
• Spectral sets in ${\mathbb{Z}}_{p^{2}qr}$ tile
• Riesz bases of exponentials and multi-tiling in finite abelian groups
• Spectral sets and tiles in ${\mathbb{Z}}_p^2\times {\mathbb{Z}}_q^2$
• Spectral structure of digit sets of self-similar tiles on ${BbbR}^1$
• On Spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem
• Classification of tile digit sets as product-forms
• The spectral eigenvalues of a class of product-form self-similar spectral measure
• Fuglede's conjecture holds in ${\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}$
• Spectral property of Cantor measures with consecutive digits
• Spectrality of generalized Sierpinski-type self-affine measures
• Spectral properties of some unions of linear spaces
• Translational absolute continuity and Fourier frames on a sum of singular measures
• The quaternary complex Hadamard matrices of orders 10, 12, and 14
• Periodicity of the spectrum of a finite union of intervals
• Turan's extremal problem on locally compact abelian groups
• Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution
• Spectrality of Self-Similar Tiles
• Algorithms for translational tiling
• Identifying Complex Hadamard Submatrices of the Fourier Matrices via Primitive Sets
• Fuglede's conjecture holds on ${\mathbb{Z}}_p^2\times {\mathbb{Z}}_q$