Some reductions of the spectral set conjecture to integers

Abstract

The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on ${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven-Meyerowitz property for finite sets of integers, introduced in \cite{CoMe99}, and we show that if the spectral sets and the tiles in ${\mathbb Z}$ satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on ${\mathbb R}^1$ are true

On the discrete Fuglede and Pompeiu problems

“…Borrowing the notation from [5] and [21], we write S − T(G) (resp. T − S(G)), if the Spectral ⇒ T ile (resp.…”

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

“…In particular, the known fact that all tiling sets of a tile and all spectra of a spectral set are periodic offers some credibility to the conjecture [LW1,IK]. Moreover, some algebraic conditions, if satisfied, are sufficient to settle the conjecture on R, although these conditions are not easy to check [DL2].…”

Fuglede’s conjecture holds on ℤ_{𝕡}²×ℤ_{𝕢}

“…The case of finite cyclic groups is particularly interesting since Dutkay and Lai [3] showed that the tile-spectral direction of Fuglede's conjecture on R holds if and only if the discrete version holds for every finite cyclic group. They also showed that if the spectral-tile direction holds on R, then it holds for every finite abelian group.…”