On product spectral sets and functional tiles in ${\mathbb{Q}}_p^d$

Abstract

This paper studies product spectral sets and functional tiles in the $p$-adic spaces ${\mathbb{Q}}_p^d$ within the framework of the {\bf product spectral set conjecture}. We establish a stability result for functional tiles under weak convergence of tiling complements, characterize product spectral pairs with product spectra, and fully resolve the conjecture for cylindric sets by proving that spectrality of a cylindric set $\Omega =B_{\gamma }(a)\times \Theta$ is equivalent to spectrality of $\Theta$.

References in corpus (8)

• The Fuglede conjecture for convex domains is true in all dimensions
• On Fuglede's conjecture and the existence of universal spectra
• The Fuglede Conjecture holds in ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$
• On the discrete Fuglede and Pompeiu problems
• Equi-distributed property and spectral set conjecture on ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$
• Fuglede's conjecture holds for cyclic groups of order $pqrs$
• Spectrality and tiling by cylindric domains
• Fuglede's conjecture on cyclic groups of order $p^{n}q$