Dense forests constructed from grids

A dense forest is a set F⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F \subset {\mathbb {R}}^n$$\end{document} with the property that for all ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} there exists a number V(ε)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(\varepsilon ) > 0$$\end{document} such that all line segments of length V(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(\varepsilon )$$\end{document} are ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-close to a point in F. The function V is called a visibility function of F. In this paper we study dense forests constructed from finite unions of translated lattices (grids). First, we provide a necessary and sufficient condition for a finite union of grids to be a dense forest in terms of the irrationality properties of the matrices defining them. This answers a question raised by Adiceam, Solomon, and Weiss (J Lond Math Soc 105: 1167–1199, 2022). To complement this, we further show that such sets generically admit effective visibility bounds in the following sense: for all η>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta > 0$$\end{document}, there exists a k∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \in {\mathbb {N}}$$\end{document} such that almost all unions of k grids are dense forests admitting a visibility function V(ε)≪ε-(n-1)-η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(\varepsilon ) \ll \varepsilon ^{-(n-1) -\eta }$$\end{document}. This is arbitrarily close to optimal in the sense that if a finite union of grids admits a visibility function V, then this function necessarily satisfies V(ε)≫ε-(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(\varepsilon ) \gg \varepsilon ^{-(n-1)}$$\end{document}. One of the main novelties of this work is that the notion of ‘almost all’ is considered with respect to several underlying measures, which are defined according to the Iwasawa decomposition of the matrices used to define the grids. In this respect, the results obtained here vastly extend those of Adiceam, Solomon, and Weiss (2022) who provided similar effective visibility bounds for a particular family of generic unimodular lattices.

Sets with this property are considered with some further restrictions. First, one may stipulate that has finite density, in the sense that lim sup Ì ½ #( 2 (0 0 0 Ì )) Ì Ò ½ Here, the symbol # denotes set cardinality. A significantly stronger restriction is uniform discreteness. This is when there is a uniform lower bound on the distance between any two distinct points in . Clearly uniform discreteness implies finite density, but the converse is false. It is a simple exercise to show that if is both a dense forest and of finite density, it can only admit visibility functions such that Î ( ) (Ò 1) . Dense forests are closely related to the Danzer Problem, which concerns whether a set Ê Ò that intersects every convex set of volume 1, called a Danzer set, exists when requiring that it has finite density.
It is known that for Ò = 2, there exists a Danzer set of finite density if and only if there exists a dense forest of finite density with visibility Î ( ) 1 [14]. However, Bambah and Woods [3] have shown that for general Ò, Danzer sets constructed from a finite union of translated lattices cannot have finite density.
The first result on the construction of dense forests was due to Peres. This can be found in a work on the question of rectifiability of curves by Bishop [4], which in fact led him to pose a version 1 of the dense forest problem. It is a planar dense forest constructed from a finite union of lattices with a visibility bound Î ( ) 4 . Although quite far from the ideal bound of Ç( 1 ), this construction has the advantage of being fully deterministic. The visibility bound of this construction was improved to Ç( 3 ) by Adiceam, Solomon, and Weiss [1]. They also proved the existence of dense forests constructed from finite unions of translated lattices admitting a visibility function Î ( ) Ò (Ò 1) for all 0. This was done by formulating a generalization of the Peres construction depending on a vector Θ Θ Θ ¾ Ê Ò 1 and then proving the visibility bound for almost all Θ Θ Θ ¾ Ê Ò 1 in the Lebesgue sense. From here-on, a translate of a lattice shall be referred to as a grid. The best known visibility bound for a deterministic construction of a dense forest is due to Tsokanos [15], namely Î ( ) (Ò 1) log( 1 ) log log( 1 ) 1+ for all 0. However, this is not uniformly discrete. It also has the peculiar property that the set depends on Î via the choice of 0. The best known bound on visibility for a uniformly discrete dense forest (in the planar case) is given by Alon [2]: 1 He allows for the centres of the 'trees' (points in the dense forest) to move as 0 varies. This is a much weaker version of what is considered a dense forest here, although the Peres construction can be easily adapted to conform to the canonical definition of a dense forest used in this paper. for some 0. It is also claimed in this paper that similar constructions may be made in general dimensions. This construction, however, is non-deterministic. Solomon and Weiss [14] also provide uniformly discrete constructions but without effective visibility bound. The only known deterministic construction of a uniformly discrete dense forest with effective visibility bound is given in [1] for the planar case, with visibility Î ( ) 5 for all 0, this also being a finite union of grids. Thus, the state of the art is fractured between the three properties: slow growth rate of the visibility function (Î ( ) 'close' to (Ò 1) ); uniform discreteness; deterministic construction. This paper examines the construction of dense forests from finite unions of grids, both from a deterministic and a metrical perspective. This work was prompted by the following problem posed in [1, §8, (2)]. This is an interesting set to study because the structure of graphene is a naturally occurring manifestation of this lattice; it can be represented as a union of two Honeycomb Lattices. An active area of scientific research is into the physical properties of what is called twisted bi-layer graphene, composed of two layers of graphene, one on top of the another (for more on this topic see [5,8]  A lattice Λ Ê Ò is associated with a matrix Å ¾ GL Ò (Ê), up to right multiplication by an element of SL Ò ( ), such that Λ = Å ¡ Ò . It turns out, however, that the dense forest property is sensitive only to a much coarser notion of equivalence. Namely, given ¾ AE, it suffices to work in the space where Ê £ denotes the multiplicative group of non-zero real numbers naturally identified with the group of homothetic matrices. This is stated more precisely in the following theorem, which further provides a necessary and sufficient condition for a finite union of grids to be a dense forest in terms of the irrationality properties of the matrices defining it. Let : GL Ò (Ê) Ë Ò denote the canonical projection.
is a dense forest.
This result establishes the dense forest property for a given union of grids without providing a visibility bound. With regards to Problem 1.1, it shows that an explicit construction may be made in the best case = 2.
The following result complements Theorem 1.1, proving that 'almost all' unions of grids are dense forests with visibility arbitrarily close to optimal as the number of grids tends to infinity. The notation := Ò 1 is used in the below, and is adopted throughout the rest of this paper.
, it produces sets uniformly close to line segments that make an angle strictly less than 2 with the ( + 1)-th axis. This does not forbid the construction dense forests; a union of + 1 rotated copies of the resulting set -specifically, the set: where, 1 +1 ¾ SO( +1) are chosen such that +1 = -produces a dense forest. The corresponding result is recorded in the following statement, where, given Ü Ü Ü ¾ Ê Ò , one lets Ì (Ü Ü Ü) := where ( ) := 2 ( + 1) 2 ¡ In the above result, only the subset Ì = Ì (Ü Ü Ü) : Ü Ü Ü ¾ Ê of the full space of upper triangular unipotent matrices is required. This is because the image of the group of matrices Ì , acting on the line Ä through 0 0 0 parallel to the ( + 1)-th axis, is the set of all lines through 0 0 0 that make an angle strictly less than 2 with Ä. On the other hand, the group of matrices SO( + 1) acting on Ä is the set of all lines through 0 0 0. Hence, Theorem 1.2 directly produces a dense forest, while Theorem 1.3 requires taking a union of rotated copies of to obtain a dense forest. The proof of Theorem 1.3 is nearly identical to that of Theorem 1.2, with only minor modifications to the arguments, and is therefore not presented here.
The proof of Theorem 1.2 arises from a focused study of density properties of linear flows on the torus, resulting in Proposition 1.4, which is of general utility. It is necessary to define some notions before stating it.
There seems to be few results in the literature on this type of question [7,11]. The above can be used to construct vectors Ù Ù Ù ¾ Ë which have optimal filling times, that is: a filling time Ì AE for some 0.
Structure of the paper. Theorem 1.1 is proved in §2, Proposition 1.4 is proved in §3, and Theorem 1.2 is proved in §4.
Acknowledgements The Author is grateful to Faustin Adiceam for his invaluable guidance throughout this project. The support of the Heilbronn Institute of Mathematical Research through the UKRI grant: Additional Funding Programme for Mathematical Sciences (EP/V521917/1) is gratefully acknowledged.
Notation Some notations shall be defined here that will remain constant throughout the paper. Several other notations will be defined later where the proper context has been provided.
• An integer Ò 2 shall be reserved for the dimension of the ambient space and is considered as fixed. As stated in the introduction, 1 is fixed to := Ò 1.
• The standard basis of Ê Ò shall be denoted by the vectors 1 Ò and the components Ü 1 Ü Ò of a vector Ü Ü Ü ¾ Ê Ò are with respect to this basis.
Since A clearly depends only on the coset in Ë Ò to which (Å Å ) belongs, it suffices to prove that A is equivalent to statement 1 of Theorem Note that the stipulation Å Ú Ú Ú ¾ Ë Ò 1 (1 ) may be lifted, since if this is not the case one may map Ú Ú Ú Ú Ú Ú Å 1 Ú Ú Ú 1 2 for each 1 . This is the negation of statement It is well-known that the closure of (2.5) is [0 1) Ò when Å 1 has rationally independent components, and a proper rational subspace otherwise (cf. [12,Chap 9]

£
The compactness argument at the end of the proof is a crucial step that precludes an explicit visibility bound.

Filling Times for Linear Flows on The Torus
In contrast to the previous section, which only required determining whether a linear flow is dense in the torus, the metrical theory necessitates quantitative bounds for the filling time of linear flows on the torus. This is the focus of Proposition 1.4, which will be proven in this section.
Given Ù Ù Ù ¾ Ë Ò 1 and AE 0, the filling time of a linear flow ∆ Ì (Ù Ù Ù) := ØÙ Ù Ù mod 1 : is the greatest lower bound on the set of Ì 0 such that (3.1) is AE-dense in [0 1) Ò . It shall be more convenient to define AE-density with respect to the supremum norm. Upon application to the construction of dense forests in subsequent sections this will have an effect which modifies the visibility functions by at most a constant factor. Assume that Ù Ù Ù ½ = Ù +1 and define The following lemma transfers the continuous flow problem to that of discrete flows.
Proof. Fix AE 0, Ù Ù Ù ¾ Ë , and identify the torus with [0 1) +1 . For each ¾ [0 1) set It is claimed that, given This provides the necessary context for the following classical transference theorem (cf. [9, Chap V, Theorem VI]) to be applied. In the below, the notation Ü denotes the greatest integer not greater than Ü ¾ Ê.
then the linear flow Proof. By Lemma 3.1, it suffices to show, given the hypotheses, that Σ Ë (Ù Ù Ù) is AE-dense in [0 1) . Proposition 3.2 implies that if, given 0, max which completes the proof.
The above provides enough machinery to prove Proposition 1.4.
Applying the triangle inequality yields Inequalities (3.5) and (3.7) establish the proposition.

The Metrical Theory
Given Ú Ú Ú ¾ Ë and ¾ (0 1), define a bi-spherical cap to be a set Ã(Ú Ú Ú ) Ë of the form: where denotes the projective distance, as defined in (1.3). Call [Ú Ú Ú] the centre of Ã, and the radius of Ã. In the proceeding metrical arguments it will be useful to have an optimal covering of Ë by bi-spherical caps (in a suitable sense), which is provided by the following lemma.
Proof. The measure of this set can be written as a product:  it can be shown that Ñ (Ê ) = Ñ ( ) for any spherical cap Ë , thereby uniquely defining Ñ as the uniform measure on Ë [10]. Therefore ( ( )) is given by the uniform measure of the bi-spherical cap Ã( ). This has a closed form (cf. [13]) which yields the approximation Ã( ) , upon denoting by ¡ the uniform measure on Ë . This implies the estimate (4.5), completing the proof.