Visible Lattice Points Research Articles

Visible lattice points along curves in higher dimensional random walks

Let k≥2 be an integer, and α=(α1,⋯,αk) with 0<α1,⋯,αk<1 and α1+⋯+αk=1, this paper concerns the generalized visibility of lattice points visited by an α-random walk in Zk. Specifically, for b=(b1,⋯,bk)∈Nk with gcd⁡(b1,⋯,bk)=1, we focus on the lattice points in the walk which are b-visible from a set of N watch-points, and we show that, almost surely, the proportion of b-visible steps is ∏p(1−N/ps(b)), where p runs over all primes and s(b):=b1+⋯+bk. This paper generalizes our previous work for the visible lattice points in higher dimensional random walks.

Visibility phenomena in hypercubes

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic themes. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in W=([0,N]∩Z)d are almost equilateral having all sides almost equal to dN/6, and the sine of the typical angle between rays from the visual spectra from the origin of W is, in the limit, equal to 7/4, as d and N/d tend to infinity. We also show that there exists an interesting number theoretic constant Λd,K, which is the limit probability of the chance that a K-polytope with vertices in the lattice W has all vertices visible from each other.

Generalized visibility of lattice points in higher dimensions

For any k≥2 and fixed b=(b1,⋯,bk)∈Nk, this paper concerns the number of integer lattice points in Zk which are b-visible from a set of N watch-points. Moreover, for N=2 and b˜=(b˜1,⋯,b˜k)∈Nk with b˜≠b, we consider the mixed visibility, that is, the number of integer lattice points which are b-visible and b˜-visible from two watch-points respectively. We prove asymptotic formulas for the number of corresponding visible lattice points. Part of our results improves a classical result of Rearick.

Generalized lattice-point visibility in ℕk

It is a well-known result that the proportion of lattice points visible from the origin is given by $\frac{1}{\zeta(2)}$, where $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika, generalized the notion of lattice point visibility by saying that for a fixed $b\in\mathbb{N}$, a lattice point $(r,s)\in\mathbb{N}^2$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f(x)=mx^b$, for some $m\in\mathbb{Q}$, between the origin and $(r,s)$. In their analysis they establish that for a fixed $b\in\mathbb{N}$, the proportion of $b$-visible lattice points is $\frac{1}{\zeta(b+1)}$, which generalizes the result in the classical lattice point visibility setting. In this short note we give an $n$-dimensional notion of $\bf{b}$-visibility that recovers the one presented by Goins et. al. in $2$-dimensions, and the classical notion in $n$-dimensions. We prove that for a fixed ${\bf{b}}=(b_1,b_2,\ldots,b_n)\in\mathbb{N}^n$ the proportion of ${\bf{b}}$-visible lattice points is given by $\frac{1}{\zeta(\sum_{i=1}^nb_i)}$. Moreover, we propose a $\bf{b}$-visibility notion for vectors $\bf{b}\in \mathbb{Q}_{>0}^n$, and we show that by imposing weak conditions on those vectors one obtains that the density of ${\bf{b}}=(\frac{b_1}{a_1},\frac{b_2}{a_2},\ldots,\frac{b_n}{a_n})\in\mathbb{Q}_{>0}^n$-visible points is $\frac{1}{\zeta(\sum_{i=1}^nb_i)}$. Finally, we give a notion of visibility for vectors $\bf{b}\in (\mathbb{Q}^{*})^n$, compatible with the previous notion, that recovers the results of Harris and Omar for $b\in \mathbb{Q}^{*}$ in $2$-dimensions; and show that the proportion of $\bf{b}$-visible points in this case only depends on the negative entries of $\bf{b}$.

Number-theoretic positive entropy shifts with small centralizer and large normalizer

AbstractHigher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

Visible lattice points along curves

This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the “visible” points and the “invisible” points in the definition of previous research. We prove asymptotic formulas for the number of lattice points in different levels of visibility.

Maximal equicontinuous generic factors and weak model sets

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice \begin{document}$ {\mathcal L}\subset G\times H $\end{document} and compact and aperiodic window \begin{document}$ W\subseteq H $\end{document} , have the maximal equicontinuous factor (MEF) \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} , if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [ 24 ]. For topological dynamical systems that possess a finite invariant measure with full support ( \begin{document}$ E $\end{document} -systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [ 33 ] and Auslander [ 2 ]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [ 34 ] that the centralizer of any \begin{document}$ {\mathcal B} $\end{document} -free dynamical system of Erdős type is trivial.

Properties of visible lattice points

This project aims to expose some properties about lattice points. More specifically, these properties are patterns on the visible lattice points (visible from the perspective of a point located at the origin of the lattice) that can be modeled by the use of the Chinese Remainder Theorem. These patterns also opens a wide spectrum of questions about the nature of prime numbers and how intimately related they are.

Visible lattice points in random walks

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability α and1−α, respectively), and starting from the origin.We show that, almost surely, the asymptotic proportion of strings of k consecutive visible lattice points visited by such an α-random walk is a certain constant ck(α), which is actually an (explicitly computable) polynomial in α of degree 2⌊(k−1)∕2⌋. For k=1, this gives that, almost surely, the asymptotic proportion of time the random walker is visible from the origin is c1(α)=6∕π2, independently of α.

Visible lattice points and the Extended Lindelöf Hypothesis

We consider the number of visible lattice points under the assumption of the Extended Lindelöf Hypothesis. We get a relation between visible lattice points and the Extended Lindelöf Hypothesis. And we also get a relation between visible lattice points over Q(−1) and the Gauss Circle Problem.

Visible lattice points and the chromatic zeta function of a graph

We study the probability that the edges of a random cycle of k vertices in the lattice $${\{1,\ldots,n\}^s}$$ do not contain more lattice points than the k vertices of the cycle. Then we introduce the chromatic zeta function of a graph to generalize this problem to other configurations induced by a given graph $${\mathcal H}$$ .

On weak model sets of extremal density

The theory of regular model sets is highly developed, but does not cover examples such as the visible lattice points, the kth power-free integers, or related systems. They belong to the class of weak model sets, where the window may have a boundary of positive measure, or even consists of boundary only. The latter phenomena are related to the topological entropy of the corresponding dynamical system and to various other unusual properties. Under a rather natural extremality assumption on the density of the weak model set, we establish its pure point diffraction nature. We derive an explicit formula that can be seen as the generalisation of the case of regular model sets. Furthermore, the corresponding natural patch frequency measure is shown to be ergodic. Since weak model sets of extremal density are generic for this measure, one obtains that the dynamical spectrum of the hull is pure point as well.

Ratio sets of random sets

We study the typical behaviour of the size of the ratio set A / A for a random subset $$A\subset \{1,\dots , n\}$$ . For example, we prove that $$|A/A|\sim \frac{2\text {Li}_2(3/4)}{\pi ^2}n^2 $$ for almost all subsets $$A\subset \{1,\dots ,n\}$$ . We also prove that the proportion of visible lattice points in the lattice $$A_1\times \cdots \times A_d$$ , where $$A_i$$ is taken at random in [1, n] with $$\mathbb P(m\in A_i)=\alpha _i$$ for any $$m\in [1,n]$$ , is asymptotic to a constant $$\mu (\alpha _1,\dots ,\alpha _d)$$ that involves the polylogarithm of order d.

On Pattern Entropy of Weak Model Sets

We study point sets arising from cut-and-project constructions. An important class is weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern entropy in terms of the volume of the window boundary in internal space. This proves a conjecture by R.V. Moody.

Ergodic Properties of Visible Lattice Points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as ℬ-free numbers with Sarnak’s conjecture on the “randomness” of the Mobius function; another is the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

Ergodic properties of visible lattice points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as ℬ-free numbers with Sarnak’s conjecture on the “randomness” of the Möbius function; another is the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

Visibility and Directions in Quasicrystals

It is well known that a positive proportion of all points in a d-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in R-d. In the present paper, we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain SL(d, R)-invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions in certain 2D cut-and-project sets. This answers some of the questions raised by Baake et al. [1].

Quotients of values of the Dedekind Eta function

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form $${\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}$$ , where A j-1 and A j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in $${\mathbb{R}^2}$$ , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.

Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n is odd and gcd ( m , n ) = 1 . Let G be the semidirect product of cyclic groups given by G = Z 8 m ⋊ Z 2 n = 〈 x , y : x 8 m = 1 , y 2 n = 1 , and yxy - 1 = x 4 m + 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are ( 0 , 0 ) , ( 4 mn 2 + 2 n , 16 m 2 n ) , ( n , 4 m ) , and ( 0 , 8 m ) . Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given by G = Z 4 m ⋊ Z 2 n = 〈 x , y : x 4 m = 1 , y 2 n = 1 , and yxy - 1 = x 2 m - 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is ( 3 m - 1 ) n + m ⌊ ( n + 1 ) / 3 ⌋ + 1 .

Visible lattice points in the sphere

The number of visible (primitive) lattice points in the sphere of radius R is well approximated by 4 π 3 ζ ( 3 ) R 3 . We consider an integral expression involving the error term E ∗ ( R ) , which leads to E ∗ ( R ) = Ω ( R ( log R ) 1 / 2 ) . This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E ∗ ( R ) . This approach to prove Ω-results seems to be new and could be useful in other problems.