Tiling, spectrality and aperiodicity of connected sets

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\mathrm{\Omega }$ tiles ${\mathbb{R}}^d$ by translations then it admits at least one periodic tiling. Fuglede’s conjecture suggests that $\mathrm{\Omega }$ admits an orthogonal basis of exponential functions if and only if it tiles ${\mathbb{R}}^d$ by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we study these conjectures for connected sets. We show that the periodic tiling conjecture, as well as both directions of Fuglede’s conjecture are false for connected sets in sufficiently high dimensions.