Splitting for Integer Tilings

Abstract

1 Introduction

Let |$A\subset{\mathbb{Z}}$| be a finite and nonempty set. We say that |$A$| tiles the integers by translations, and call it a finite tile, if there is a set |$T\subset{\mathbb{Z}}$| such that each |$n\in{\mathbb{Z}}$| has a unique representation |$n=a+t$| with |$a\in A$| and |$t\in T$|⁠. It is well known [32] that any such tiling must be periodic, so that there exists a |$M\in{\mathbb{N}}$| such that |$T=B\oplus M{\mathbb{Z}}$| for some finite set |$B\subset{\mathbb{Z}}$|⁠. We then have |$|A|\,|B|=M$|⁠, so that |$A\oplus B={\mathbb{Z}}_{M}$| is a factorization of the cyclic group |${\mathbb{Z}}_{M}$|⁠.

By translational invariance, we may assume that |$A,B\subset \{0,1,\dots \}$|⁠. The mask polynomials of |$A$| and |$B$| are

In this language, |$A\oplus B={\mathbb{Z}}_{M}$| is equivalent to

Recall that the |$s$|-th cyclotomic polynomial |$\Phi _{s}(X)$| is the minimal polynomial of |$e^{2\pi i/s}$|⁠. The identity

allows us to rephrase (1.1) as

Since |$\Phi _{s}$| are irreducible, each |$\Phi _{s}(X)$| with |$s|M$| must divide at least one of |$A(X)$| and |$B(X)$|⁠.

The Coven–Meyerowitz theorem [3] is as follows.

We do not know whether all finite tiles satisfy (T2). The statement that this is true has become known in the literature as the Coven–Meyerowitz conjecture and is generally considered to be the main open question in the theory of integer tilings. The progress to date includes our work in [23, 24], leading to Theorem 1.3 in the odd case and partial results for general tilings, as well as mild extensions of the Coven–Meyerowitz theorem in [29, 34, 36], and proofs of (T2) for tilings of certain special types [4, 21]. There is also significant interest in other tiling questions, for example, translational tilings of |${\mathbb{Z}}^{n}$| and |${\mathbb{R}}^{n}$| [1, 11, 13, 14] and tilings of the real line by a function [18].

The main result of this paper is the following theorem. For |$N\in{\mathbb{N}}$|⁠, we will use the notation

Theorem 1.2 is a natural application of splitting, a new method developed in this article and described briefly below. The proof is much shorter and simpler than those in [23] and [24]. Splitting is also an important part of the proof of Theorem 1.3 below. In [24], we proved Theorem 1.3 under the additional assumption that |$M$| is odd. In the companion paper [25] we use a combination of splitting and the methods of [23, 24] to extend the same result to the even case, thus completing the proof of the theorem.

In order to apply Theorem 1.2, we need to have a number |$M^{\prime}$| such that (T2) holds for both tiles in any |$M^{\prime}$|-periodic tiling. Such choices of |$M^{\prime}$| are provided by Theorems 1.1, 1.3, and [23, Corollary 6.2]. Corollary 1.4 below states the most general assumptions on |$M$| under which our work here, combined with [23–25], yields (T2) for both tiles in any tiling |$A\oplus B={\mathbb{Z}}_{M}$|⁠.

The two special cases below use Theorem 1.2 together with Theorems 1.1 and 1.3.

We note the application to Fuglede’s conjecture [9], which (in its original formulation) states that a set |$\Omega \subset{\mathbb{R}}^{n}$| of positive |$n$|-dimensional Lebesgue measure tiles |${\mathbb{R}}^{n}$| by translations if and only if the space |$L^{2}(\Omega )$| admits an orthogonal basis of exponential functions. A set with the latter property is called spectral. The conjecture is known to be false, in its full generality, in dimensions 3 and higher [7, 8, 19, 20, 31, 35]. However, there are important special cases in which the conjecture was confirmed [12, 15, 28] and the finite abelian group analogue of the conjecture is currently a very active area of research; see, for example, [6, 16, 17, 29, 30, 34].

In dimension 1, the problem is still open in both directions, and the “tiling implies spectrum” direction hinges on proving (T2) for all finite tiles ([22, 26, 27]; see also [5] for an overview of the problem and an investigation of the converse direction). Our Corollary 1.4, combined with [22, Theorem 1.5] and [23, Corollary 6.2], yields the following result.

As mentioned above, our arguments are based on splitting (Section 4). Given a fiber in |${\mathbb{Z}}_{M}$| (an arithmetic progression of step |$M/p$| and length |$p$| for some prime divisor |$p|M$|⁠), we ask which elements of |$A$| and |$B$| tile the elements of that progression. It is not difficult to prove that, for each fiber separately, these elements must follow a certain “splitting” pattern. If these splitting patterns are uniform for all fibers corresponding to some prime factor |$p$|⁠, and if this uniformity persists under dilations of the sets |$A$| and |$B$|⁠, we are able to apply a tiling reduction developed in [23, Section 6.2].

If the assumption (1.2) holds, it is relatively easy both to prove such uniformity and to set up an inductive argument proving (T2). For other classes of tilings, further work is needed. Towards this end, in Section 7 we prove several partial results on splitting consistency on grids that are needed in the proof of Theorem 1.3 in [25].

An additional goal of this paper is to revisit and simplify some of our results from [23, 24]. In Section 3, we provide a simple proof of Theorem 3.7 based on the dilation theorems of Sands and Tijdeman, and a new combinatorial interpretation of concepts (such as saturating sets) derived from that theorem. In Section 5, we give a new formulation of the slab reduction from [23] in terms of splitting. This offers an easier way to verify the criteria of the reduction (see, e.g., [25, Section 9]), and supplies a key part of our proof of Theorem 1.2.

2 Notation and Preliminaries

We assume that |$M=p_{1}^{n_{1}}\dots p_{K}^{n_{K}}$|⁠, where |$p_{1},\dots ,p_{K}$| are distinct primes and |$n_{1},\dots ,n_{K}\in{\mathbb{N}}$|⁠. If |$A\subset{\mathbb{Z}}_{M}$|⁠, its mask polynomial is |$A(X)=\sum _{a\in A}X^{a}$|⁠. We use the convolution notation |$A*B$| for the weighted sumset of |$A$| and |$B$| (interpreted as a multiset if necessary), so that |$(A*B)(X)=A(X)B(X)$|⁠. If one of the sets is a singleton, say |$A=\{x\}$|⁠, we write |$x*B=\{x\}*B$|⁠. The direct sum notation |$A\oplus B$| is reserved for tilings, so that |$A\oplus B={\mathbb{Z}}_{M}$| means that |$A,B\subset{\mathbb{Z}}_{M}$| are both sets and |$A(X)B(X)=\frac{X^{M}-1}{X-1}$| mod |$X^{M}-1$|⁠.

The Chinese Remainder Theorem identifies the cyclic group |${\mathbb{Z}}_{M}$| with |${\mathbb{Z}}_{p_{1}^{n_{1}}}\oplus{\mathbb{Z}}_{p_{2}^{n_{2}}}\oplus \ldots \oplus{\mathbb{Z}}_{p_{K}^{n_{K}}}$|⁠. This allows us to interpret |${\mathbb{Z}}_{M}$| as a |$K$|-dimensional lattice, periodic in each direction, with an explicit coordinate system defined as follows. For |$j\in \{1,\dots ,K\}$|⁠, let |$M_{j}: = M/p_{j}^{n_{j}}= \prod _{i\neq j} p_{i}^{n_{i}}$|⁠. Then each |$x\in{\mathbb{Z}}_{M}$| has a unique representation

For |$D|M$|⁠, a |$D$|-grid in |${\mathbb{Z}}_{M}$| is a set of the form

for some |$x\in{\mathbb{Z}}_{M}$|⁠. We note a few important special cases.

• •

  A line through |$x\in{\mathbb{Z}}_{M}$| in the |$p_\nu $| direction is the set |$\ell _\nu (x):= \Lambda (x,M_\nu )$|⁠.

• •

  A plane through |$x\in{\mathbb{Z}}_{M}$| perpendicular to the |$p_\nu $| direction, on the scale |$M_\nu p_\nu ^{\alpha _\nu }$|⁠, is the set |$\Pi (x,p_\nu ^{\alpha _\nu }):=\Lambda (x,p_\nu ^{\alpha _\nu }).$|

• •
  A fiber in the |$p_\nu $| direction is a set of the form |$x*F_\nu $|⁠, where |$x\in{\mathbb{Z}}_{M}$| and

  $$ \begin{align*} & F_\nu=\{0,M/p_\nu,2M/p_\nu,\dots,(p_\nu-1)M/p_\nu\}. \end{align*} $$

  Thus, |$x*F_\nu =\Lambda (x,M/p_\nu )$|⁠.

A set |$A\subset{\mathbb{Z}}_{M}$| is fibered in the |$p_{j}$| direction if there is a subset |$A^{\prime}\subset A$| such that |$A=A^{\prime}*F_{j}$|⁠. Since in this paper we only consider fibers at the top scale |$M$|⁠, we use the terms “fiber” and “fibered” instead of “|$M$|-fiber” and “|$M$|-fibered” as in [23–25].

3 Divisor Isometries and the Box Product

3.1 Divisor isometries

We continue to assume that |$M=p_{1}^{n_{1}}\dots p_{K}^{n_{K}}$|⁠, where |$p_{1},\dots ,p_{K}$| are distinct primes. The divisor set of |$A\subset{\mathbb{Z}}_{M}$| is the set

Our analysis here is motivated in part by the fundamental theorems of Sands [33] and Tijdeman [37], which we now state.

Examples of divisor isometries include translations |$\tau _{c}(x)=x-c$| for any |$c\in{\mathbb{Z}}_{M}$| and invertible dilations |$\psi _{r}(x)= rx$| for any |$r\in R$|⁠, where

We note that the composition of any number of translations and dilations is always a linear transformation. An example of a divisor isometry that is not a linear mapping is provided by a plane exchange mapping, defined as follows. Let |$c,c^{\prime}\in{\mathbb{Z}}_{M}$| satisfy |$(c-c^{\prime},M)=M_{i}p_{i}^{\alpha -1}$| for some |$i\in \{1,\dots ,K\}$| and |$0< \alpha \leq n_{i}$|⁠. Then,

Lemma 3.4 below is an immediate consequence of Theorem 3.1.

We use |$\phi $| to denote the Euler totient function: if |$n=\prod _{i=1}^{L} q_{i}^{\beta _{i}}$|⁠, where |$q_{1},\dots ,q_{L}$| are distinct primes and |$\beta _{i}\in{\mathbb{N}}$|⁠, then |$ \phi (n)= \prod _{i=1}^{L} (q_{i}-1)q_{i}^{\beta _{i}-1}. $|

A brief discussion of the relationship between Theorem 3.1, Theorem 3.2, and Lemma 3.4 is in order. The special case of Tijdeman’s theorem with |$r\in R$| was first proved by Sands [33, Theorem 2]. Sands used this to prove Theorem 3.1 [33, Theorem 3]. Lemma 3.4 uses Theorem 3.1 to extend [33, Theorem 2] to more general divisor isometries.

Theorem 3.2 extends [33, Theorem 2] in a different direction. Instead of assuming as in [33, Theorem 2] that |$r\in R$|⁠, Tijdeman only assumes that |$r$| is relatively prime to |$|A|$|⁠. This is a significantly weaker assumption, since the mapping |$x\to rx$| with |$(r,|A|)=1$| need not be a divisor isometry if |$|B|$| has prime factors that |$|A|$| does not have. Coven and Meyerowitz [3] used the full strength of Theorem 3.2 to prove that if |$|A|$| tiles the integers, then it must in fact tile a cyclic group |${\mathbb{Z}}_{M}$| for some |$M$| with the same prime factors as |$|A|$|⁠.

3.2 A combinatorial interpretation of the box product

Following [23, 24] we define for |$x\in{\mathbb{Z}}_{M}$|

If |$X\subset{\mathbb{Z}}_{M}$| and |$x\in{\mathbb{Z}}_{M}$|⁠, we write |${\mathbb{A}}_{m}[x|X] = \# \{a\in A\cap X: \ (x-a,M)=m\}$|⁠.

For |$A,B\subset{\mathbb{Z}}_{M}$|⁠, we define their box product

The proof of Theorem 3.7 in [10, 23] uses discrete harmonic analysis. Below, we present an alternative combinatorial proof, based on averaging over dilations |$\psi _{r}$| with |$r\in R$|⁠.

3.3 Saturating sets

Let |$A\oplus B= {\mathbb{Z}}_{M}$|⁠, and |$x,y\in{\mathbb{Z}}_{M}$|⁠. We define

and similarly for |$B_{y,x}$| with |$A$| and |$B$| interchanged. These are the sets that saturate the box product in (3.3), in the sense that

We also define the saturating set for |$x$| with respect to |$A$|⁠,

with |$B_{y}$| defined similarly.

We note an alternative description of saturating sets, based on dilations and closely related to the alternative proof of Theorem 3.7 presented above.

4 Splitting

4.1 Splitting for fibers

We continue to assume that |$M=p_{1}^{n_{1}}\dots p_{K}^{n_{K}}$| and |$A\oplus B={\mathbb{Z}}_{M}$|⁠.

These are the sets of elements of |$A$| and |$B$| that tile |$Z$|⁠. Note that |$\Sigma _{A}(Z)$| depends on both |$A$| and |$B$|⁠; whenever more than one tiling complement of |$A$| is being considered, we will identify the relevant tiling explicitly.

An extension to lower scales is easy and could be added with only minimal additional effort, but will not be needed here.

Splitting is closely related to saturating sets. Our dilation-based reinterpretation of the latter shows them to be the sets of elements of |$A$| and |$B$| that tile a fixed element |$z$| of |${\mathbb{Z}}_{M}$| in tilings |$A\oplus rB$|⁠, where |$r$| ranges over an appropriate set of dilations. Splitting does not take dilations into account; however, instead of tiling just one element of |${\mathbb{Z}}_{M}$|⁠, we tile an entire arithmetic progression at the same time.

4.2 Splitting uniformity

We now focus on the consequences of |$A$|-uniform splitting parity.

We do not know whether the converse of Corollary 4.9 holds, that is, whether |$\Phi _{p_{i}^{n_{i}}}|A$| implies that the tiling |$A\oplus B={\mathbb{Z}}_{M}$| has |$A$|-uniform |$(B,A)$| splitting parity in the |$p_{i}$| direction.

5 Tiling Reductions

The slab reduction in Theorem 5.1 and Corollary 5.2 was introduced in [23] and used in the proof of (T2) for 3 prime factors in [24].

If a tiling |$A\oplus B={\mathbb{Z}}_{M}$| satisfies the conditions of Theorem 5.1, we can reduce proving (T2) for |$A$| and |$B$| to proving it for the tilings |$A^{\prime}_{p_{i}}\oplus B={\mathbb{Z}}_{M/p_{i}}$| defined above [23, Corollary 6.7]. (In [23], the first assumption of Corollary 6.7 is stated for all |$i$|⁠; however, the proof only requires it to hold for the value of |$i$| for which [23, Theorem 6.5] is applied.) Corollary 5.2 below follows by iterating this procedure until (T2) is known to hold.

Clearly, if |$A$| and |$B$| satisfy (5.1) then the same holds for |$A$| and |$rB$| for all |$r\in R$|⁠. Thus, |$A$| and |$B$| satisfy the slab reduction conditions if and only if the same is true for |$A$| and |$rB$| with any |$r\in R$|⁠.

In the rest of this section, we aim to connect the slab reduction with uniform splitting parity (Definition 4.6) in a fixed direction. We will also need a formulation of uniform splitting in terms of box products. Recall that the box product notation |${\mathbb{A}}[x]$| and |$\langle{\mathbb{A}}[x],{\mathbb{B}}[y]\rangle $| was defined at the beginning of Section 3.2. For a set |$X\subset{\mathbb{Z}}_{M}$|⁠, we also define

We proved in [23, Section 9.2] that whenever |$x\in a*F_{i}$| for some |$i\in \{1,\dots ,K\}$| and |$a\in A$|⁠, then |$A_{x}\subset (\Pi (a,p_{i}^{n_{i}})\cup \Pi (x,p_{i}^{n_{i}}))$|⁠. In general, we do not know how to prove that |$A_{x}$| is in fact contained in only one of the above planes. However, the following special case is of interest.

Lemma 5.4 below establishes a rigorous relationship between the slab reduction, uniform splitting, and uniform box product splitting in the sense of Definition 5.3. Additionally, Lemma 5.4 provides a direct link between the slab reduction and [23, Conjecture 9.4], while Corollary 5.5 (i) may be viewed as a step towards resolving Conjecture 9.2 in [23].

6 Proof of Theorem 1.2 and Corollary 1.4

6.1 Proof of Theorem 1.2

Let |$M=p_{1}^{n_{1}}\dots p_{K}^{n_{K}}$|⁠, where |$p_{1},\dots ,p_{K}$| are distinct primes and |$n_{1},\dots ,n_{K}\in{\mathbb{N}}$|⁠. Assume that |$A\oplus B={\mathbb{Z}}_{M}$|⁠. We first prove the theorem under the stronger assumption that

where |$M_{1}=p_{2}^{n_{2}}\dots p_{K}^{n_{K}}$|⁠. We first claim that (6.1) implies uniform splitting in the |$p_{1}$| direction. Assume towards contradiction that |$z*F_{1}$| splits with parity |$(A,B)$| and |$z^{\prime}*F_{1}$| splits with parity |$(B,A)$| for some |$z,z^{\prime}\in{\mathbb{Z}}_{M}$|⁠. We have |$|\Sigma _{B}(z*F_{1})|=p_{1}>M_{1}$|⁠, so that at least two elements of |$\Sigma _{B}(z*F_{1})$| lie on one line in the |$p_{1}$| direction, with |$M/p_{1}\in{\mathrm{Div}}(B)$|⁠. Similarly, the |$(B,A)$| splitting parity implies that |$M/p_{1}\in{\mathrm{Div}}(A)$|⁠. However, by divisor exclusion it is not possible for both of these to hold.

Assume therefore that the tiling has uniform |$(B,A)$| splitting parity, with |$M/p_{1}\in{\mathrm{Div}}(A)$|⁠. By the same argument as above, all tilings |$A\oplus rB={\mathbb{Z}}_{M}$| with |$r\in R$| also have uniform |$(B,A)$| splitting parity. It follows from Lemma 5.4 that the tiling |$A\oplus B={\mathbb{Z}}_{M}$| satisfy the slab reduction conditions in Theorem 5.1. By Corollary 5.2, to prove that |$A$| and |$B$| satisfy (T2), it suffices to prove that (T2) holds for both sets in any tiling |$A^{\prime}\oplus B^{\prime}={\mathbb{Z}}_{M^{\prime}}$|⁠, where |$M^{\prime}=M/p_{1}$|⁠. Iterating the procedure, we eventually reduce the question to proving (T2) for all tilings of period |$M_{1}$|⁠. But at that point, the assumption on |$M_{1}$| implies the conclusion.

For the full strength of the theorem, we need the proposition below.

We return to the proof of Theorem 1.2. Assume that (1.2) holds, so that

We proceed by induction in |$M$|⁠. To initialize, if |$M$| has at most 2 distinct prime factors, then the conclusion follows from the Coven–Meyerowitz theorem. Assume now that the theorem holds for all |$\widetilde{M}|M$| with |$\widetilde{M}\neq M$|⁠, and let |$A\oplus B={\mathbb{Z}}_{M}$| be a tiling. If there exists some |$i\in \{1,\dots ,K\}$| such that |$p_{i}\nmid |A|$| or |$p_{i}\nmid |B|$|⁠, we may apply the subgroup reduction in [3, Lemma 2.5] (see also Theorem 6.1 and Corollary 6.2 in [23]) to reduce proving (T2) for |$A$| and |$B$| to our inductive assumption on tilings of |${\mathbb{Z}}_{M/p_{i}}$|⁠.

We may therefore assume that |$p_{i}$| divides both |$|A|$| and |$|B|$| for each |$i$|⁠. This implies that

and similarly for |$|B|$|⁠. By divisor exclusion, at least one of |$M/p_{1}\not \in{\mathrm{Div}}(A)$| and |$M/p_{1}\not \in{\mathrm{Div}}(B)$| must hold. Assume without loss of generality that the latter holds. Applying Proposition 6.1 with |$i=1$|⁠, we see that the tiling has uniform |$(B,A)$| splitting parity in the |$p_{1}$| direction, and that |$M/p_{1}\in{\mathrm{Div}}(A)$|⁠. The rest of the proof is as above: by Lemma 5.4, |$A$| and |$B$| satisfy the slab reduction conditions in Theorem 5.1. By Corollary 5.2, it suffices to prove (T2) for both tiles in any tiling of period |$\widetilde{M}=M/p_{1}$|⁠. But this again follows from our inductive assumption.

6.2 Proof of Corollary 1.4

Lemma 6.2 is based on the same argument as [23, Corollary 6.2] (see also [4, 34, 36]), but we modify the statement slightly to fit the present context.

We now prove Corollary 1.4. Let

where |$p_{1},\dots ,p_{K},q_{1},\dots ,q_{L}$| are distinct primes and the assumptions of Corollary 1.4 hold.

Each of the primes |$p_{4},\dots ,p_{K}$|⁠, as well as |$p_{3}$| if (1.3) holds, appears in the factorization of |$M$| only once. Hence, each of these primes may divide at most one of |$|A|$| and |$|B|$| in any tiling |$A\oplus B={\mathbb{Z}}_{M}$| with |$M|M^{*}$|⁠. By Lemma 6.2, it suffices to prove the corollary with

which we now assume. Using (1.5), and applying Theorem 1.2 inductively, we can also remove all of the |$q_{j}$| primes, starting with |$q_{L}$| and working back to |$q_{1}$|⁠. This leaves us with proving the corollary with

Let |$M|M^{*}$| and |$A\oplus B={\mathbb{Z}}_{M}$|⁠. If |$M=N_{0} =p_{1}^{2}p_{2}^{2}p_{3}^{2}$| and |$|A|=|B|=p_{1}p_{2}p_{3}$|⁠, then both |$A$| and |$B$| satisfy (T2) by Theorem 1.3. If only two of the primes |$p_{1},p_{2},p_{3}$| divide |$M$|⁠, then |$A$| and |$B$| satisfy (T2) by Theorem 1.1. In all other cases, there is an |$i\in \{1,2,3\}$| such that |$p_{i}$| divides at most one of |$|A|$| and |$|B|$|⁠. We then apply Lemma 6.2 to pass from a tiling of period |$M$| to tiling of period |$M/p_{i}$|⁠. Continuing this procedure, we eventually get to the point where Theorem 1.1 applies.

7 Splitting Consistency

Our work in Sections 5 and 6 shows that uniform splitting offers a viable path to proving the Coven–Meyerowitz conjecture. The goal of this section is to place the first few building blocks in the systematic study of uniform splitting.

Partial results towards establishing uniform splitting might involve proving that splitting is uniform on certain subgrids of |${\mathbb{Z}}_{M}$|⁠. We are also interested in splitting consistency—extensions of Lemma 4.3 saying that the elements of |$A$| and |$B$| that tile certain grids larger than a fiber must come from cosets of proper subgroups of |${\mathbb{Z}}_{M}$|⁠. In Section 7.1, we prove such a result for 2-dimensional grids; a result of this type for 3-dimensional grids, but with additional assumptions, is given in Lemma 7.6.

In Section 7.2, we study tiling structure and splitting under the assumption that one of the sets |$A$| and |$B$| has fibered intersections with all |$D(M)$|-grids. To provide some context, we describe briefly our approach to the case |$M=p_{i}^{n_{i}}p_{j}^{n_{j}}p_{k}^{n_{k}}$|⁠. In [24, 25], given a tiling |$A\oplus B={\mathbb{Z}}_{M}$|⁠, we analyze the set whose mask polynomial is divisible by |$\Phi _{M}$| (say, |$A$|⁠). This divisibility condition places constraints on the structure of |$A$| on any fixed |$D(M)$|-grid. Broadly speaking, on any fixed grid |$A$| may either be written as a disjoint union of fibers in some fixed direction, or else it must have one out of several “unfibered” structures we have identified explicitly. We say that |$A$| is fibered on |$D(M)$|-grids if the former holds on every grid |$\Lambda (a,D(M))$| for every |$a\in A$|⁠. Our results in Section 7.2 are an important part of the argument in the fibered grids case in [25].

7.1 General results

We start with two results on splitting consistency on subgrids. In addition to applications in [25], Lemma 7.1 has been used in the proof of Lemma 6.4 in [2].

7.2 Splitting for fibered grids

In this section, we assume the following.

Assumption (F): We have |$A\oplus B={\mathbb{Z}}_{M}$|⁠, where |$M=p_{i}^{n_{i}}p_{j}^{n_{j}} p_{k}^{n_{k}}$| has three distinct prime factors. Furthermore, |$A$| is fibered on all |$D(M)$|-grids.

In particular, Assumption (F) implies that |$\Phi _{M}|A$|⁠, since |$\Phi _{M}\mid x*F_\nu $| for each |$x\in{\mathbb{Z}}_{M}$| and |$\nu \in \{i,j,k\}$|⁠.

Let |${\mathcal{I}}$| be the set of elements of |$A$| that belong to a fiber in the |$p_{i}$| direction, that is,

The sets |${\mathcal{J}}$| and |${\mathcal{K}}$| are defined similarly. The assumption (F) implies that |$A={\mathcal{I}}\cup{\mathcal{J}}\cup{\mathcal{K}}$|⁠. In general, this does not have to be a disjoint union and an element of |$A$| may belong to two or all three of these sets.

We will write |$D=D(M)$| for short. We also define the direction assignment function |$\kappa :A\to \{i,j,k\}$| as follows. Write |$A=\bigcup _{x\in \{1,\dots ,M/D\}} A\cap \Lambda (x,D)$|⁠. By (F), for each |$x\in \{1,\dots ,M/D\}$| such that |$A\cap \Lambda (x,D)$| is nonempty, there exists |$\nu (x)\in \{i,j,k\}$| such that |$A\cap \Lambda (x,D)$| is fibered in the |$p_{\nu (x)}$| direction. (If there exists more than one such |$\nu $|⁠, we choose one arbitrarily and fix that choice.) We then let |$\kappa (a):=\nu (x)$| for all |$a\in A\cap \Lambda (x,D)$|⁠.

With this definition, we have |$\kappa (a)=\kappa (a^{\prime})$| for all |$a,a^{\prime}\in A$| such that |$D|(a-a^{\prime})$|⁠. We also define

Observe that, for any |$a,a^{\prime}\in A$|⁠, the fibers |$F(a)$| and |$F(a^{\prime})$| are either identical or disjoint. Indeed, suppose that |$a^{\prime\prime}\in F(a)\cap F(a^{\prime})$|⁠; then |$a,a^{\prime}$| belong to the same grid |$\Lambda (a^{\prime\prime},D)$|⁠, so that |$\kappa (a)=\kappa (a^{\prime})$| by definition, and |$F(a)=F(a^{\prime})$|⁠.

We will prove that every |$D$|-grid is tiled by fibers in at most 2 directions, with the “stratified” structure described in Lemma 7.4. We then prove a few results concerning the localization of |$\Sigma _{A}(\Lambda )$|⁠, in the same spirit as Lemma 7.1, but with stronger conclusions since we now have the fibering assumption at our disposal.

The geometric intuition for the next result is provided by tilings of a 3-dimensional space by rectangular columns. Define |$R_{i},R_{j},R_{k}\subset{\mathbb{R}}^{3}$| by

Then any tiling of |${\mathbb{R}}^{3}$| by translated, pairwise disjoint (up to sets of measure zero) copies of |$R_{i},R_{j},R_{k}$| may use translated copies of at most two of these sets. Moreover, the tiling can be stratified into slabs of thickness 1, with each slab tiled by translates of just one of |$R_{i},R_{j},R_{k}$|⁠.

We briefly discuss splitting parity and the slab reduction for fibered grids. If |$A\oplus B={\mathbb{Z}}_{M}$|⁠, where |$M$| has at most 3 prime factors, our expectation is that the condition of the slab reductions are satisfied for at least one of |$A$| and |$B$| in some direction [23, Conjecture 9.1]. By Lemma 5.4, this means uniform splitting parity in that direction for all tilings |$A\oplus rB={\mathbb{Z}}_{M}$| with |$r\in R$|⁠. An intermediate result in this direction might establish uniform splitting parity on certain subgrids of |${\mathbb{Z}}_{M}$|⁠. For example, the following is easy to prove.

It is likely that, at least for |$n_{i}=n_{j}=2$|⁠, Lemma 7.6 can be proved using our current methods without the assumption that (7.10) holds. However, since this extension is not needed in the proofs of our main results, we do not attempt it here.

Funding

The first author is supported by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant 22R80520). The second author is supported by the Israel Science Foundation (Grant 607/21).

Acknowledgments

The authors would like to thank the anonymous referee for their useful remarks and suggestions.

Communicated by Rick Kenyon

References