On the central ball in a translation invariant involutive field

TL;DR

This work characterizes a novel metric on $\ ime \mathbb{Z}^2$ induced by two translation-invariant involutions, resulting in a partition of the plane into parabolas with stairway-like paths. The parabolic-taxicab distance $d_{pc}$ combines fast stair-step motion along these paths with traditional grid moves, yielding intricate ball shapes whose boundaries lie between the lines $y=x\pm r$ and exhibit parity-driven pulsations. The authors give a complete description of balls centered at the origin, derive exact formulas for their size and boundary points, and prove a previously conjectured area formula for these parabolic-taxicab balls. The results illuminate geometric and combinatorial structures arising from the interplay of parabolic partitions and taxicab movement, with potential implications for related lattice-geometry problems.

Abstract

The iterated composition of two operators, both of which are involutions and translation invariant, partitions the set of lattice points in the plane into an infinite sequence of discrete parabolas. Each such parabola contains an associated stairway-like path connecting certain points on it, induced by the alternating application of the aforementioned operators. Any two lattice points in the plane can be connected by paths along the square grid composed of steps either on these stairways or towards taxicab neighbors. This leads to the notion of the parabolic-taxicab distance between two lattice points, obtained as the minimum number of steps of this kind needed to reach one point from the other. In this paper, we describe patterns generated by points on paths of bounded parabolic-taxicab length and provide a complete description of the balls centered at the origin. In particular, we prove an earlier conjecture on the area of these balls.

Figures (6)

• Figure 1: In the image on the left-side $\mathscr{B}_{\mathrm{pc}}(O,13)$ is shown, and in the image on the right-side the three consecutive borders of the balls with radii $10, 11$, and $12$ are shown. The cardinalities are: $\#\mathscr{B}_{\mathrm{pc}}(O,13)=1987$, $\#\partial \mathscr{B}_{\mathrm{pc}}(O,10)= 242$, $\#\partial \mathscr{B}_{\mathrm{pc}}(O,11)= 294$, and $\#\partial \mathscr{B}_{\mathrm{pc}}(O,12)=350$.
• Figure 2: Highlighted are the sets whose projections on the first coordinate are equal to $\mathcal{S}(r,c)$ and $\mathcal{S}_{\raisebox{-1pt}{-}}(r,c)$ for $r=9$, $c =\pm 9$ (left) and $r=9$, $c =\pm 5$ (right). Thus, we have: $\mathcal{S}_{\raisebox{-1pt}{-}}(9,9)=[-9,-1]\cap\mathbb{Z}$, $\mathcal{S}(9,9)=[-9,36]\cap\mathbb{Z}$; $\mathcal{S}(9,-9)=[0,45]\cap\mathbb{Z}$, $\mathcal{S}_{\raisebox{-1pt}{-}}(9,-9)=\emptyset$; $\mathcal{S}_{\raisebox{-1pt}{-}}(9,5)=[-18,-12]\cap\mathbb{Z}$, $\mathcal{S}(9,5)=([-18,-12]\cup[17,23])\cap\mathbb{Z}$; $\mathcal{S}_{\raisebox{-1pt}{-}}(22,-5)=[-13,-7]\cap\mathbb{Z}$, $\mathcal{S}(9,-5)=([-13,-7]\cup[22,28])\cap\mathbb{Z}$.
• Figure 3: The real plane partitioned by parabolas $x+y+2m=(x-y)^2$, with $m\in\mathbb{Z}$ (left). Steps of length $1$ as measured by the parabolic-taxicab distance (right) by applying $L'$, then $L"$, (or vice-versa) and continuing to alternate. The indicated points have coordinates $(T_k,T_{k+1})$ and $(T_{k+1},T_{k})$, where $T_k=\hbox{$\dbinom{k+1}{2}$}$ are the triangular numbers.
• Figure 4: The boundary of $\mathscr{B}_{\mathrm{pc}}(O,6)$. Three points $P$ marked with $\ast$ are chosen as follows: $P=(-6,-6)$ (left), $P=(2,8)$ (middle), $P=(21,15)$ (right). All points $Q\in\partial\mathscr{B}_{\mathrm{pc}}(O,6)$ are shown in colors indicating the parabolic-taxicab distance from $P$ to $Q$.
• Figure 5: The set of lattice points situated at distance at most $4$ from the points in the set $\mathcal{C} = \{(3,-3), (-3,3)\}$ (left) and $\mathcal{C} = \{(0,0),(4,-4), (-4,4)\}$ (right).

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof