Invariant sub-tiling structure of measurable tilings of tori in dimensions $d \geq 3$

Grebík, Greenfeld, Rozhoň and Tao proved that every measurable tiling $F \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$ of a torus by finitely many translates of a single measurable tile can be linearly deformed into a tiling with rational shifts. In dimension $d = 1$ the tiling is always rational, and in $d = 2$ the deformation is rigid: all velocities are parallel to one vector and the tiling splits into sub-tilings invariant under the corresponding one-parameter flow. In dimensions $d \geq 3$ this strong rigidity provably fails, and the paper leaves open whether anything survives: no example is known of a measurable tiling of $\mathbb{T}^d$, $d \geq 3$, for which no non-zero velocity $v$ admits a proper non-empty sub-family of shifts $F' \subset F$ whose partial tiling $F' \oplus A$ is invariant under the flow $x \mapsto x + tv$; it is likewise unknown whether every such tiling satisfies even the weaker analogue of the two-dimensional structure in which shifts move along piecewise-linear paths. The question is open.

Background

A measurable tiling problem takes a measure space $X$ carrying a measure-preserving action of an abelian group $G$, a finite set of shifts $F \subset G$, and a measurable tile $A \subset X$, and asks when the translates $f \cdot A$, $f \in F$, partition $X$ up to null sets — written $F \oplus A =_{\mathrm{a.e.}} X$, where $=_{\mathrm{a.e.}}$ means equality up to a set of measure zero. This is the measurable counterpart of the classical theory of translational tilings of the integer lattice, in which a finite set $F \subset \mathbb{Z}^d$ tiles $\mathbb{Z}^d$ when $F \oplus A = \mathbb{Z}^d$ for some set $A$ of translation vectors. A central theme there is the periodic tiling conjecture (PTC), ascribed notably to Lagarias and Wang: if a finite tile $F$ tiles $\mathbb{Z}^d$, must it also admit a periodic tiling, i.e., one whose tiling set is a finite union of cosets of a finite-index sublattice of $\mathbb{Z}^d$. The PTC is true in $d = 1$ (elementary) and $d = 2$ (Bhattacharya, via ergodic theory; quantitatively by Greenfeld–Tao), and was famously disproved by Greenfeld–Tao for sufficiently large $d$, so the boundary between 'structured' and 'wild' tilings is known to lie somewhere in between; the case $d = 3$ remains open.

The source paper of this problem — 'Measurable tilings by abelian group actions' by Grebík, Greenfeld, Rozhoň and Tao — extends the discrete structure theory to the measurable category. Its central tool is a measurable dilation lemma: $F \oplus A =_{\mathrm{a.e.}} X$ implies $F^r \oplus A =_{\mathrm{a.e.}} X$ for a large family of integer dilations $r$ of the shift set ($F^r$ denoting the pointwise $r$-th power in multiplicative notation for the group operation). Dilation forces long-range correlations between far-apart pieces of a tiling, and the paper builds from it a structure theory with two flagship applications. First, a complete classification of the finite tiles that admit a factor-of-iid tiling of a finitely generated abelian group $G$ (Theorem 1.5): a factor-of-iid tiling is a random tiling produced as a measurable, shift-equivariant function of independent identically distributed random labels on the group elements — informally, a tiling generated by local randomness — and the classification shows that the only such tiles are those contained in a single coset of the finite-torsion part of $G$ (for $G = \mathbb{Z}^d$, only singletons). Second, a structure theorem for measurable tilings of the torus (Theorem 1.7), which is where the present problem lives.

The torus $\mathbb{T}^d = \mathbb{R}^d / \mathbb{Z}^d$ carries the translation action of $\mathbb{R}^d$ (with Haar measure, i.e., Lebesgue measure), so a measurable tiling $F \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$ uses a finite shift set $F = \{f_1, \dots, f_n\} \subset \mathbb{R}^d$ and one measurable tile $A \subset \mathbb{T}^d$ whose translates cover the torus exactly once almost everywhere. Such a tiling is rational if all shift differences $f_i - f_j$ have rational coordinates. Theorem 1.7 shows that every measurable torus tiling can be linearly deformed into a rational one: there are rational positions $f_i^0 \in \mathbb{Q}^d$ such that, setting the velocities $v_i = f_i - f_i^0$ and letting the shift set slide as $F^t = \{f_i^0 + t v_i\}$, one has $F^t \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$ for every $t \in \mathbb{R}$. A measurable set $S \subset \mathbb{T}^d$ is called $\mathbb{R}v$-invariant for a velocity $v \in \mathbb{R}^d$ if $tv + S =_{\mathrm{a.e.}} S$ for all $t \in \mathbb{R}$, i.e., $S$ is invariant under the one-parameter flow $x \mapsto x + tv$; on the torus, the orbit closures of this flow are cosets of a subtorus (a consequence of Kronecker's theorem on simultaneous rational approximation).

The rigidity of the deformation depends on the dimension. For $d = 1$, every measurable tiling is rational, which settled the one-dimensional case of a conjecture of Conley–Grebík–Pikhurko from their work on divisibility of spheres with measurable pieces. For $d = 2$ (after translating so that $0 \in F$), all velocities are parallel to a single integer vector $v$, and $F$ partitions into velocity classes $F_j$ whose partial tilings $F_j \oplus A = \{f + a : f \in F_j, a \in A\}$ are each $\mathbb{R}v$-invariant: the tiling decomposes into sub-tilings individually invariant under one common flow. For $d \geq 3$ this strong conclusion is false — the paper gives an explicit tiling of $\mathbb{T}^3$ by the cube $A = [0,1/2]^3$ with eight shifts, some involving irrational coordinates, for which grouping the shifts by velocity provably does not yield invariant sub-tilings. What remains open in $d \geq 3$ is whether any vestige of the rigidity survives, and two known facts delimit the question. First, by an argument of Szabó from his reduction of Keller's cube-tiling conjecture, if the tile $A$ is a cube then every tiling by it does satisfy a weaker analogue of the two-dimensional structure, in which the constant velocities are replaced by piecewise-linear paths. Second, if several distinct tiles are allowed, the authors construct three measurable tiles in $\mathbb{T}^3$ (bounded by sine-graph surfaces) forming a tiling whose motion cannot be explained by sliding the tiles individually or by translating the whole configuration — so full non-rigidity is achievable once multiple tiles are permitted. The open question is the single-tile case, and it is Remark 5.5 of the source paper.

Problem Statement

Let $d \geq 3$ and let $\mathbb{T}^d = \mathbb{R}^d / \mathbb{Z}^d$ be the $d$-dimensional torus with Haar (Lebesgue) measure. Let $F = \{f_1, \dots, f_n\} \subset \mathbb{R}^d$ be a finite shift set with $n \geq 2$ and $A \subsetneq \mathbb{T}^d$ a measurable tile such that $F \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$, i.e., the translates $f_i + A$ partition $\mathbb{T}^d$ up to a null set. Recall that $S \subset \mathbb{T}^d$ is $\mathbb{R}v$-invariant if $tv + S =_{\mathrm{a.e.}} S$ for every $t \in \mathbb{R}$. This is Remark 5.5 of Grebík–Greenfeld–Rozhoň–Tao; determine which of the following two surviving forms of the two-dimensional rigidity (Theorem 1.7(ii) of that paper) holds in dimensions $d \geq 3$. (1) Does there exist a measurable tiling $F \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$, for some $d \geq 3$, that does not satisfy the weaker analogue of Theorem 1.7(ii) — that is, for which there exist no piecewise-linear paths $v_f : [0,1] \to \mathbb{R}^d$ ($f \in F$) with $F^t = \{v_f(t) : f \in F\}$ defined analogously as in Theorem 1.7, so that the paths deform a rational tiling shift set $F^0 = \{v_f(0) : f \in F\} \subset \mathbb{Q}^d$ at $t = 0$ into the original shift set $F$ at $t = 1$ (thus $v_f(1) = f$ for each $f \in F$), and with the following two properties: first $F^t \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$ for all $t \in [0,1]$, and second, for every velocity $v \in \mathbb{R}^d$ and every $t$ at which the derivatives $v_f'(t)$ exist, the sub-family $F^t_{(v)} = \{f \in F : v_f'(t) = v\}$ satisfies that $F^t_{(v)} \oplus A$ is $\mathbb{R}v$-invariant? (2) More weakly, does there exist a measurable tiling $F \oplus A =_{\mathrm{a.e.}} \mathbb{T}^d$, for some $d \geq 3$, such that there is no velocity $0 \neq v \in \mathbb{R}^d$ and no proper non-empty sub-family $\emptyset \neq F' \subsetneq F$ for which $F' \oplus A$ is $\mathbb{R}v$-invariant? It is known that cube tiles always satisfy the weaker analogue in (1) (an argument of Szabó), that tilings in $d \leq 2$ are always structured, and that fully non-rigid tilings of $\mathbb{T}^3$ do exist once more than one tile is allowed.

References

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