A counterexample to the periodic tiling conjecture (announcement)

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R^d}$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z^2} \times G_0$ for some finite abelian $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations

Undecidability of translational monotilings

In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in $\mathbb{Z}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually $\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2\times G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in $\mathbb{Z}^d$, $d\geq 3$.Comment: 44 pages, 10 figures, typos correcte

A counterexample to the periodic tiling conjecture

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.Comment: 50 pages, 13 figures. Minor changes and additions of new reference

Tiling, spectrality and aperiodicity of connected sets

Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that $\Omega$ 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.Comment: 20 pages, 8 figure

An uncountable ergodic Roth theorem and applications

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This extends a previous result of Bergelson, McCutcheon and Zhang. Using this uncountable Roth theorem, we establish the following two additional results. [(i)] We establish a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. [(ii)] We establish a uniform bound on the lower Banach density of the set of double recurrence times along all $\Gamma$-systems, where $\Gamma$ is any group in a class of uniformly amenable groups. As a special case, we obtain this uniformity over all $\mathbb{Z}$-systems, and our result seems to be novel already in this particular case. Our uncountable Roth theorem is crucial in the proof of both of these results.Comment: 34 pages, [v2]: typos corrected, [v3]: improved presentation following referee's feedback, title and abstract change