2.5 Density of sets in the lattice $\mathbb{Z}^{2}$

One of the aims of this report is the investigation of the density of intersections of integer circles in $\mathbb{Z}^{2}$ .

Definition 2.5.1.

Let $D$ be some set, and let $\mathcal{D}=(D_{n})$ be a filtration of $D$ . We say that $\mathcal{D}$ is an exhaustive filtration of $D$ if for every positive integer $n$ we have ${D}_{n}\subset D_{n+1}$ and

\bigcup\limits_{n=1}^{\infty}D_{n}=D.

Definition 2.5.2.

Let $\mathcal{D}=(D_{n})$ be an exhaustive filtration of non-empty finite subsets of an at most countable set $D$ . Let $A$ be a subset of $D$ . A density of the elements of $A$ with respect to the exhaustive filtration $\mathcal{D}$ is the number

\#_{\mathcal{D}}A=\lim\limits_{n\to\infty}\frac{|A\cap D_{n}|}{|D_{n}|},

if the limit exists, where $|S|$ is the number of elements in $S$ .

This notion of density is considered with respect to these exhaustive filtrations of subset.

Definition 2.5.3.

Let us consider the filtration $\mathcal{I}=(I_{n})$ , where

I_{n}=\big{(}[-n,n]\times[-n,n]\big{)}\cap\mathbb{Z}^{2}.

Example 2.5.4.

Recall that the density of a single integer circle is given by the formula

\#_{\mathcal{I}}S_{r}(O)=\frac{6}{(r\pi)^{2}}.

Definition 2.5.5.

The butterfly filtration generated by unit integer vector $(a,b)$ with $a\neq 0$ and $b\neq 0$ , denoted $\mathcal{B}(a,b)=(B_{n}(a,b))$ is the exhaustive filtration of $\mathbb{Z}^{2}$ where $B_{n}\subset\mathbb{Z}$ is the set of points bounded by lines $y=bn$ , $y=bx/an$ , $y=-bx/an$ and $y=-bn$ .

Remark 2.5.6.

To form an invariant filtration on $\mathbb{Z}^{2}$ under integer affine transformations, let the butterfly filtration generated by angle $\operatorname{ltan}()$ be the butterfly filtration generated by () and all integer transformations of it.

2.5 Density of sets in the lattice ℤ2\mathbb{Z}^{2}