Rotation on the digital plane

The digital plane is a lattice whose elements are points with integer coordinates, the so called lattice points. The values of continuous functions can be represented only approximatively, rounding its computed value to a lattice point. Rounding is a mapping $q\;:\;{\mathbb{R}}^{2}\mapsto{\mathbb{Z}}^{2}$. The discrete variant of the function $f\;:\;{\mathbb{R}}^{2}\mapsto{\mathbb{R}}^{2}$ is $q\circ f$. Of course there are plenty discrete variants of $f$. One of the most studied function of the plane is the rotation, which is a $2\times 2$ real matrix with eigenvalues $\cos\varphi+i\sin\varphi$ and $\cos\varphi-i\sin\varphi$. Let us denote it by $Q_{\varphi}$. There exists an invertible real matrix $Q$ such that $Q_{\varphi}=QA_{\varphi}Q^{-1}$ with

For $(x,y)\in{\mathbb{R}}^{2}$ the sequence of points $\{f^{n}(x,y)\}_{n=0}^{\infty}$ is called an orbit¹¹1Some author calls this sequence trajectory. of $f$ generated by $(x,y)$. The orbits of $Q_{\varphi}$ generated by any non-zero points lie on an ellipse, those of $A_{\varphi}$ on the unit circle. Plainly $Q_{\varphi}$ is a bijective mapping on ${\mathbb{R}}^{2}$, but usually not bijective on ${\mathbb{Z}}^{2}$. Combining it with a rounding, which results the function $q\circ Q_{\varphi}$ we obtain a mapping of ${\mathbb{Z}}^{2}\mapsto{\mathbb{Z}}^{2}$, which is a discrete rotation in wide sense. Many interesting and hard questions appear: is a discrete rotation injective, surjective or bijective? How are its orbits? All orbits of $Q_{\varphi}$ are bounded, but this is not at all clear for its discrete variants. The investigation of such questions have long tradition, see the early example [6].

FLOOR²²2Rounds a real point to the nearest lattice point down on the left., ROUND³³3Stands for rounding off a vector to the nearest node of the lattice. and TRUNC ⁴⁴4Denotes the coordinate-wise truncation of the fractional part of a vector towards the zero point. are eminent examples of rounding functions. Discretizing the rotation with them the resulted mappings ${\mathbb{Z}}^{2}\mapsto{\mathbb{Z}}^{2}$ are more or less different. Kozyakin et al [9] gave a good overview on the results concerning discretized rotations, especially on $ROUND\circ A_{\varphi}$ and $TRUNC\circ A_{\varphi}$. Diamond et al [7] proved that if $\varphi\not=k\frac{\pi}{2},k\in{\mathbb{Z}}$ then all orbits of $TRUNC\circ A_{\varphi}$ eventually gets into the zero point. The situation is very different with $ROUND\circ A_{\varphi}$. Kozyakin et al [9] proved among others that if the rotation angle $\varphi$ is such that the rows of all the nonnegative powers of the matrix $A_{\varphi}$ are rationally independent then the density of lattice points with empty full preimages is positive. They used measure theoretic approach, which allow to prove much more general results too. For other probabilistic results on discrete rotations we refer to [6, 13, 14, 8].

In former investigations of the second author with different coauthors [1, 2, 3, 4] a kind of discrete rotation appears as a natural generalization of positional number systems. It was $FLOOR\circ B_{\varphi}$ with

We come back to this function later, but before we discuss some properties of the FLOOR function. It commutes with the additive group of translations of ${\mathbb{Z}}^{2}$ and the full preimage of zero is $[0,1[\times[0,1[$, which is Jordan measurable, thus FLOOR, like ROUND, is a quantizer in the sense of [9]. Hence the discretized rotation FLOOR$\circ A_{\varphi}$ has similar properties as ROUND$\circ A_{\varphi}$. This holds, among others, for the above mentioned property of preimages.

In this note we prove under the natural assumption $\varphi\not=k\frac{\pi}{2}$ that the function $FLOOR\circ A_{\varphi}$ is neither surjective nor injective. More precisely we prove lower density estimates for the sets of lattice points, which are the image of two lattice points as well as of lattice points, which have no preimages. It turns out, see Theorems 3.1 and 4.2, that these densities are positive except when $\sin\varphi\not=\pm\cos\varphi+r,r\in{\mathbb{Q}}$. This means that the number of such lattice points lying in a box symmetric to the origin and of side length $2M+1$ is $O(M^{2})$. In contrast in the exceptional case this number is only $O(M)$. In Section 6 we indicate that the same results hold to $ROUND\circ A_{\varphi}$ too. We use in the proof elementary results of uniform distribution theory and properties of primitive Pythagorean triplets. Our results are more precise than those in [9].

There are discrete rotations which are bijective. Trivial examples are $FLOOR\circ A_{\varphi}$ with $\varphi=k\frac{\pi}{2},k\in{\mathbb{Z}}$. More interesting are the functions $FLOOR\circ B_{\varphi},0\leq\varphi<2\pi$. Reeve-Black and Vivaldi [12] claim that a generic discrete rotation is neither injective nor surjective. Our results justify this claim for the function $FLOOR\circ A_{\varphi}$. To prove similar characterization for $FLOOR\circ Q_{\varphi}$ is a challenging problem. We expect that apart the previous examples only the transpose of $B_{\varphi}$ lie in the exceptional set.

Despite many efforts and interesting results, we have deterministic knowledge only on the orbits of $FLOOR\circ B_{\varphi}$. For the eleven values $2\cos\varphi=\lambda=0,\pm 1,(\pm 1\pm\sqrt{5})/2,\pm\sqrt{2},\pm\sqrt{3}$ all orbits are periodic, see [10, 2, 3]. Generally it was proved by Akiyama and Pethő [4] that for any $\varphi$ there are infinitely many periodic orbits. This is still far from the conjecture that all orbits are periodic see [2].

The matrix $A_{\varphi}$ through $A_{\varphi}\cdot(a,b),(a,b)\in{\mathbb{R}}^{2}$ ⁵⁵5To be precise we had to write $A_{\varphi}\cdot(a,b)^{T}$ instead of $A_{\varphi}\cdot(a,b)$, where $(a,b)^{T}$ denote the transpose of the vector $(a,b)$, i.e., a column vector. As in the article we should do this often, and from the context it will be clear whether the actual vector is a row or a column vector, we avoid this extra notation. induces a linear mapping on ${\mathbb{R}}^{2}$, which we will denote by $A_{\varphi}$ too.

FLOOR and integer part $\lfloor.\rfloor$ are the same functions, in the sequel we will use the later. To simplify our notation we define $r_{\varphi}\;:\;{\mathbb{Z}}^{2}\mapsto{\mathbb{Z}}^{2}$ by

We also use the fractional part function, i.e, $\{x\}=x-\lfloor x\rfloor$. Both functions will be applied coordinate wise to the points of the real vector spaces. We use the same notation to these extended functions. Let $U=[0,1[\times[0,1[$ and $\bar{U}=[0,1]\times[0,1]$, then obviously $r_{\varphi}(a,b)=(x,y)\in{\mathbb{Z}}^{2}$ if and only if $A_{\varphi}\cdot(a,b)=(x,y)+u$ for some $u\in U$. The third equivalent expression is $\{A_{\varphi}\cdot(a,b)\}=A_{\varphi}\cdot(a,b)-r_{\varphi}(a,b)$.

In order to prove our main results we need some tools from uniform distribution theory. Let ${\bf a}=(a_{1},\ldots,a_{n}),{\bf b}=(b_{1},\ldots,b_{n})\in{\mathbb{R}}^{n}$ be such that $0\leq a_{j}<b_{j}\leq 1,\;j=1,\ldots,n$ then we set $B_{{\bf a,b}}=[a_{1},b_{1}[\times\ldots\times[a_{n},b_{n}[$. This is a box with side lengths $b_{1}-a_{1},\ldots,b_{n}-a_{n}$, whose volume is plainly $\prod_{j=1}^{n}(b_{j}-a_{j})$. For a sequence of $n$-dimensional real vectors $X=({\bf x}_{m})$ set

where $|S|$ denotes the cardinality of the set $S$.

The sequence $X$ is called uniformly distributed modulo $1$, shortly uniformly distributed if

holds for all ${\bf a,b}\in{\mathbb{R}}^{n}$ with the above property. Notice that if $X$ is uniformly distributed then there exist for any $B_{{\bf a,b}}$ a constant $c=c_{\bf a,b}>0$ such that

holds for all large enough $N$. Indeed, one may set $c=\prod_{j=1}^{n}(b_{j}-a_{j})-\varepsilon$ for some $\varepsilon>0$.

The following theorem is a bit modified version of Theorem I, p.64. of Cassels [5] and it plays a crucial role in this paper.

Now we formulate the main lemma of this paper.

Before stating the main result of this section we introduce a notation: $T_{\varphi}(M)$ denotes the number of $(x,y)\in{\mathbb{Z}}^{2}$ such that $|x|,|y|\leq M$ and $(x,y)$ is the image by $r_{\varphi}$ of two different grid points. If $\varphi=k\frac{\pi}{2},k\in{\mathbb{Z}}$ then $r_{\varphi}=A_{\varphi}$, thus it is bijective, thus $T_{\varphi}(M)=0$. Otherwise, we prove that $T_{\varphi}(M)$ tends to infinity.

In the last section we proved that the digital rotation is usually not injective. Now we prove that usually it is not surjective either. To achieve our goal we need an elementary geometric lemma. To state and prove it we introduce some notation. For a point $(a,b)\in\mathbb{R}^{2}$ put $T_{(a,b)}=(a,b)+U$ and $\bar{T}_{(a,b)}=(a,b)+\bar{U}$, where $+$ means here translation. The squares $T_{(a,b)},\;(a,b)\in{\mathbb{Z}}^{2}$ are disjoint and their union cover $\mathbb{R}^{2}$. As $A_{\varphi}$ is a rotation the sets $A_{\varphi}(T_{(a,b)})$ are squares too with the same properties. Thus there exists for any $(n,m)\in\mathbb{Z}^{2}$ unique $(a,b)\in{\mathbb{Z}}^{2}$ such that $(n,m)\in A_{\varphi}(T_{(a,b)})$.