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Aaron Meyerowitz
Aaron Meyerowitz
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A periodic $(n,k)$ coloring can be modified to give a non-periodic $(mn+\ell,mk+\ell)$ coloring for any non-negative $m,k$$m,\ell$ not both $0$ (this uses some comments to improve my previous answer, although there seem to be even better answers given) ). This is a modification of Anthony's construction without need of probabilities. This allows the periodic $(3,2)$-colorings to give non-periodic $(a+b,a)$ colorings for any $a \ge 2b$ (except $(a+b,a)=(3,2)$)

Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and let the new colors be $c_i$ for $1 \le i \le m$ and $1 \le c \le n$ along with new colors $-1,-2,\cdots,-\ell.$ Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Go through the zones and in each do one of the following (being sure to do each thing infinitely often in each possible way)

  • change eacheach color $c$ to $c_i$ (for an $1 \le i \le m$ constant on that zone.)

  • change everything to $-j$ (for a $1 \le j \le \ell$, constant on that zone)

The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single zone, then several full periods in the next zone etc.

A periodic $(n,k)$ coloring can be modified to give a non-periodic $(mn+\ell,mk+\ell)$ coloring for any non-negative $m,k$ not both $0$ (this uses some comments to improve my previous answer, although there seem to be even better answers given) ). This is a modification of Anthony's construction without need of probabilities. This allows the periodic $(3,2)$-colorings to give non-periodic $(a+b,a)$ colorings for any $a \ge 2b$ (except $(a+b,a)=(3,2)$)

Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and let the new colors be $c_i$ for $1 \le i \le m$ along with new colors $-1,-2,\cdots,-\ell.$ Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Go through the zones and in each do one of the following (being sure to do each thing infinitely often in each possible way)

  • change each color $c$ to $c_i$ (for an $1 \le i \le m$ constant on that zone.)

  • change everything to $-j$ (for a $1 \le j \le \ell$, constant on that zone)

The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single zone, then several full periods in the next zone etc.

A periodic $(n,k)$ coloring can be modified to give a non-periodic $(mn+\ell,mk+\ell)$ coloring for any non-negative $m,\ell$ not both $0$ (this uses some comments to improve my previous answer, although there seem to be even better answers given) ). This is a modification of Anthony's construction without need of probabilities. This allows the periodic $(3,2)$-colorings to give non-periodic $(a+b,a)$ colorings for any $a \ge 2b$ (except $(a+b,a)=(3,2)$)

Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and let the new colors be $c_i$ for $1 \le i \le m$ and $1 \le c \le n$ along with new colors $-1,-2,\cdots,-\ell.$ Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Go through the zones and in each do one of the following (being sure to do each thing infinitely often in each possible way)

  • change each color $c$ to $c_i$ (for an $1 \le i \le m$ constant on that zone.)

  • change everything to $-j$ (for a $1 \le j \le \ell$, constant on that zone)

The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single zone, then several full periods in the next zone etc.

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Aaron Meyerowitz
Aaron Meyerowitz
  • 30.1k
  • 1
  • 48
  • 104

A periodic $(n,k)$ coloring can be modified to give a non-periodic $(2n,2k)$$(mn+\ell,mk+\ell)$ coloring for any non-negative $m,k$ not both $0$ (this uses some comments to improve my previous answer, although there seem to be even better answers given) ). This is a modification of Anthony's construction without need of probabilities. This allows the periodic $(3,2)$-colorings to give non-periodic $(a+b,a)$ colorings for any $a \ge 2b$ (except $(a+b,a)=(3,2)$)

Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and allow $n$ morelet the new colors $\{{-1,-2,,\cdots,-n\}}$be (This is just$c_i$ for ease of description). Start$1 \le i \le m$ along with new colors $-1,-2,\cdots,-\ell.$ Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Alternate makingGo through the colors positivezones and negative according toin each do one of the zone.following (being sure to do each thing infinitely often in each possible way)

  • change each color $c$ to $c_i$ (for an $1 \le i \le m$ constant on that zone.)

  • change everything to $-j$ (for a $1 \le j \le \ell$, constant on that zone)

The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single positive zone zone, then several full periods in a single negative zone.

I am not convinced that there are periodic ones, but I am not convinced there are not. The first construction with diagonal matrices (if I understand correctly) has some lines of slope $-1$ with just $1,2.$ A line of slope $-1$ going through a point labelled $0$ would seem to have all points labelled $0.$ I did not look at the other constructions, and may misunderstand the first onenext zone etc.

A periodic $(n,k)$ coloring can be modified to give a non-periodic $(2n,2k)$ coloring. This is a modification of Anthony's construction without need of probabilities. Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and allow $n$ more colors $\{{-1,-2,,\cdots,-n\}}$ (This is just for ease of description). Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Alternate making the colors positive and negative according to the zone. The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single positive zone, then several in a single negative zone.

I am not convinced that there are periodic ones, but I am not convinced there are not. The first construction with diagonal matrices (if I understand correctly) has some lines of slope $-1$ with just $1,2.$ A line of slope $-1$ going through a point labelled $0$ would seem to have all points labelled $0.$ I did not look at the other constructions, and may misunderstand the first one.

A periodic $(n,k)$ coloring can be modified to give a non-periodic $(mn+\ell,mk+\ell)$ coloring for any non-negative $m,k$ not both $0$ (this uses some comments to improve my previous answer, although there seem to be even better answers given) ). This is a modification of Anthony's construction without need of probabilities. This allows the periodic $(3,2)$-colorings to give non-periodic $(a+b,a)$ colorings for any $a \ge 2b$ (except $(a+b,a)=(3,2)$)

Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and let the new colors be $c_i$ for $1 \le i \le m$ along with new colors $-1,-2,\cdots,-\ell.$ Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Go through the zones and in each do one of the following (being sure to do each thing infinitely often in each possible way)

  • change each color $c$ to $c_i$ (for an $1 \le i \le m$ constant on that zone.)

  • change everything to $-j$ (for a $1 \le j \le \ell$, constant on that zone)

The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single zone, then several full periods in the next zone etc.

Source Link
Aaron Meyerowitz
Aaron Meyerowitz
  • 30.1k
  • 1
  • 48
  • 104

A periodic $(n,k)$ coloring can be modified to give a non-periodic $(2n,2k)$ coloring. This is a modification of Anthony's construction without need of probabilities. Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and allow $n$ more colors $\{{-1,-2,,\cdots,-n\}}$ (This is just for ease of description). Start with the $(n,k)$ coloring and draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Alternate making the colors positive and negative according to the zone. The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single positive zone, then several in a single negative zone.

I am not convinced that there are periodic ones, but I am not convinced there are not. The first construction with diagonal matrices (if I understand correctly) has some lines of slope $-1$ with just $1,2.$ A line of slope $-1$ going through a point labelled $0$ would seem to have all points labelled $0.$ I did not look at the other constructions, and may misunderstand the first one.