A lonely weak tile

TL;DR

This paper reviews weak tiling in the context of spectral sets and provides a counterexample of a set that weakly tiles its complement but is neither spectral nor a proper tile, challenging previous assumptions.

Contribution

It introduces a novel example of a set that weakly tiles its complement without being spectral or a proper tile, addressing an open question in the field.

Findings

Provided a counterexample set that weakly tiles its complement

Showed the set is neither spectral nor a proper tile

Clarified the relationship between weak tiling and spectral sets

Abstract

The notion of weak tiling was a key ingredient in the proof of Fuglede's spectral set conjecture for convex bodies \cite{conv}, due to the fact that every spectral set tiles its complement weakly with a suitable Borel measure. In this paper we review the concept of weak tiling, and answer a question raised in \cite{weak} by giving an example of a set $T$ which tiles its complement weakly, but $T$ is neither spectral, nor a proper tile.