Some reductions of the spectral set conjecture to integers

Original · EN

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 R¹, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on Zₙ, Z and R¹ 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 R¹ 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 CoMe99, and we show that if the spectral sets and the tiles in Z satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on R¹ are true.