Some reductions of the spectral set conjecture to integers

By Dorin Ervin Dutkay, Chun-Kit Lai. Mathematical Proceedings of the Cambridge Philosophical Society, 2013

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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 [1], 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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0e293d122e956076b6880a48d4f9ce9a08af5b21

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In Applied Mathematics

  1. Xiaoye Fu
  2. Xinggang He
  3. Ka-Sing Lau
  4. Xin-Rong Dai
  5. Li-Xiang An
  6. Chun-Kit Lai
  7. Qiyu Sun
  8. Alexander Grigorʼyan

In Discrete Mathematics and Combinatorics

  1. Xinggang He
  2. Ka-Sing Lau
  3. Xiaoye Fu

In Geometry and Topology

  1. Xinggang He
  2. Ka-Sing Lau

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