Complex Hadamard matrices and the Spectral Set Conjecture

TL;DR

This paper proves the spectral implies tile direction of the Spectral Set Conjecture for small sets in finite groups and extends the results to Euclidean spaces, also identifying counterexamples in low dimensions.

Contribution

It establishes the spectral implies tile direction for sets of size up to 5 in finite groups and shows the conjecture fails in dimensions as low as 3.

Findings

Spectral implies tile holds for sets of size ≤ 5 in finite Abelian groups.

Counterexamples to the conjecture exist in dimension 3.

The paper discusses computational complexity issues related to spectral and tiling properties.

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 $\Z^d$ 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 $\Z_3^5$, and this observation quickly led to the failure of the Spectral Set Conjecture in $\R^5$ (Tao), and subsequently in $\R^4$ (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.