Related papers: Simultaneous visibility in the integer lattice
Let $K$ be a number field with ring of integers $\mathcal{O}$. Two lattice points ${\bf x, y}\in \mathcal{O}^m$ with $m\geq 2$ are said to be visible from one another if $\gcd((x_i-y_i),\ldots, (x_m-y_m))=\mathcal{O}$, where $(x_i-y_i)$ is…
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…
A lattice point $\vec x=(x_1,\dots,x_n)\in\mathbb Z^{n}$ is said to be visible if the line segment between $\vec x$ and the origin contains no other lattice point. In this paper, we compute the asymptotic density of visible lattice points…
We consider the number of visible lattice points under the assumption of the Extended Lindel\"{o}f Hypothesis. We get a relation between visible lattice points and the Extended Lindel\"{o}f Hypothesis. And we also get a relation between…
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 $\mathscr…
The relative density of visible points of the integer lattice $\mathbb{Z}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann's zeta function. In this paper we prove that the relative density of visible points in the…
For a fixed $b\in\mathbb{N}=\{1,2,3,\ldots\}$ we say that a point $(r,s)$ in the integer lattice $\mathbb{Z} \times \mathbb{Z}$ is $b$-visible from the origin if it lies on the graph of a power function $f(x)=ax^b$ with $a\in\mathbb{Q}$ and…
We look at the average sum of the Euler's phi function $\phi{(n)}$ and it's relation with the visibility of a point from the origin.We show that $\forall{\hspace{0.05in}{k} \ge{1}},k\in\mathbb{N},\exists$ a $k$$\times$$k$ grid in the 2D…
Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the…
We prove that the set of visible points of any lattice of dimension at least 2 has pure point diffraction spectrum, and we determine the diffraction spectrum explicitly. This settles previous speculation on the exact nature of the…
We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with…
We say a lattice point $X=(x_1,\ldots,x_m)$ is visible from the origin, if $\gcd(x_1,...,x_m)=1$. In other word, there are no other lattice point on the line segment from the origin $O$ to $X$. From J.E. Nymann's result, we know that the…
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,…
We comment on the set of visible points of a lattice and its Fourier transform, thus continuing and generalizing previous work by Schroeder and Mosseri. A closed formula in terms of Dirichlet series is obtained for the Bragg part of the…
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 $\mathbb{R}^d$. In the present paper we prove an analogous…
In this paper, for any integer $k\geq 2$, we study the distribution of the visible lattice points in certain generalized P\'{o}lya's walk on $\mathbb{Z}^k$: perturbed P\'{o}lya's walk and twisted P\'{o}lya's walk. For the first case, we…
The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its…
We study points in cut-and-project sets which are visible from the origin, continuing a direction of inquiry initiated in [6,14] where the asymptotic density of visible points was investigated. We establish an error bound for the density of…
We revisit the visible points of a lattice in Euclidean $n$-space together with their generalisations, the $k$th-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice…
Recently, the notion of visibility from the origin has been generalized by viewing lattice points through curved lines of sights, where the family of curves considered are $y=mx^k$, $k\in\mathbb{N}$. In this note, we generalize the notion…