Spiral Delone sets and three distance theorem

TL;DR

This paper characterizes when a constant angle progression on the Fermat spiral forms a Delone set, linking it to the property of the angle being badly approximable.

Contribution

It establishes a precise condition relating the angle's Diophantine approximation properties to the Delone set formation on the Fermat spiral.

Findings

Constant angle progression forms a Delone set iff the angle is badly approximable.

Provides a characterization connecting number theory and geometric point sets.

Enhances understanding of spatial distributions on spirals.

Abstract

We show that a constant angle progression on the Fermat spiral forms a Delone set if and only if its angle is badly approximable.

Equations

$$X(f,\alpha )=\{f(n)\mathbf{e}(n\alpha )|n\in \mathbb{N}\}$$

$$\bigcup _{z\in X(f,\alpha )}B(x,r)\supset \mathbb{C}.$$

$$\bigcup _{z\in K[n]}B(z,r)\supset B(0,f(n)-r)$$

$$\bigcup _{z\in K[n]}B(z,s)\subset B(0,f(n)+s)$$

$$X(\sqrt{t},\alpha )=\{\sqrt{n}\mathbf{e}(n\alpha )|n\in \mathbb{N}\}.$$

$$r>1/2\text{ and }s<2$$

$$q|q\alpha -p|\ge C$$

$$\frac{1}{B+2}\le C\le \frac{1}{B},$$

$$\{n\alpha (\mathrm{mod}1)|n\in \{0,1,\dots ,N\}\}.$$

$${\eta }_{k-1}=a_{k+1}{\eta }_k+{\eta }_{k+1},0\le {\eta }_{k+1}<{\eta }_k$$

$$\frac{{\eta }_{k-1}}{{\eta }_k}=a_{k+1}+\frac{1}{\frac{{\eta }_k}{{\eta }_{k+1}}},$$

$$\alpha =[0;a_1,a_2,a_3,\dots ]=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots }}}}$$

$$p_{k+1}=a_{k+1}{p}_k+p_{k-1},q_{k+1}=a_{k+1}{q}_k+q_{k-1}$$

$${\eta }_k>\frac{1}{q_{k+1}+q_k}>\frac{1}{2q_{k+1}},$$

$$N=s{q}_k+q_{k-1}+r$$

$$1<\frac{{\eta }_{k-1}-(s-1){\eta }_k}{{\eta }_k}\le \frac{{\eta }_{k-1}}{{\eta }_k}\le 1+a_{k+1}$$

$$1<\frac{{\eta }_{k-1}-(s-1){\eta }_k}{{\eta }_{k-1}-s{\eta }_k}=1+\frac{{\eta }_k}{{\eta }_{k-1}-s{\eta }_k}\le 1+\frac{{\eta }_k}{{\eta }_{k+1}}\le 2+a_{k+2}.$$

$$|{\eta }_{k-1}/{\eta }_k-a_{k+1}|<1.$$

$$B-1\le R\le B+2.$$

$$4\Vert x-y\Vert \le |\mathbf{e}(x)-\mathbf{e}(y)|\le 2\pi \Vert x-y\Vert$$

$$m-3r\sqrt{m}\le n\le m+3r\sqrt{m},\Vert n\alpha -\beta \Vert \le \frac{r}{2\sqrt{m}}.$$

$$r>|\sqrt{n}\mathbf{e}(n\alpha )-\sqrt{m}\mathbf{e}(\beta )|\ge |\sqrt{n}-\sqrt{m}|$$

$$|n-m|\le (\sqrt{n}+\sqrt{m})r\le (2\sqrt{m}+r)r\le 3\sqrt{m}r.$$

$$r$$

$$\frac{r}{2\sqrt{m}}>\Vert n\alpha -\beta \Vert .$$

$$|\sqrt{n}\mathbf{e}(n\alpha )-z|$$

$$n\alpha (\mathrm{mod}1)\text{ with }n\in \mathbb{Z}\cap [m-3r\sqrt{m},m+3r\sqrt{m}]$$

$$6r\sqrt{m}-1<N+1\le 6r\sqrt{m}+1.$$

$$m-3r\sqrt{m}\le n\le m+3r\sqrt{m},\Vert n\alpha -\beta \Vert \le \frac{R}{2(6r\sqrt{m}-1)}\le \frac{r}{2\sqrt{m}}$$

$$r\ge 6\max \left\{\sqrt{\frac{B+2}{5}},1\right\}$$

Full text

In botany, phyllotaxis stands for angular patterns of plant stems or leaves, which tends to be correlated with Fibonacci numbers and the golden mean. In this short note, we study a related fundamental problem on the spiral patterns, which is not addressed before. Let $X$ be a subset of $\mathbb{R}^{2}$ which is identified with the complex plane $\mathbb{C}$. Denote by $B(x,r)$ the open ball of radius $r$ centered at $x$. $X$ is $r$-relatively dense if there exists $r>0$ that for any $x\in\mathbb{C}$, $B(x,r)\cap X\neq\emptyset$ holds. $X$ is $s$-uniformly discrete if there exists $s>0$ that for any $x\in\mathbb{C}$ we have $\text{Card}(B(x,s)\cap X)\leq 1$. $X$ is a $(r,s)$-Delone set if it is both $r$-relatively dense and $s$-uniformly discrete. It is clear that $r$-relatively dense implies $r^{\prime}$-relatively dense if $r\leq r^{\prime}$, and $s$-uniformly discrete implies $s^{\prime}$-uniformly discrete if $s\geq s^{\prime}$, and $r\geq s$ holds for a $(r,s)$-Delone set. We omit the prefixes $r$-, $s$- and $(r,s)$- if we are only interested in the existence of $r$ and/or $s$.

Denote by $\mathop{\mathbf{e}}\nolimits(z)=e^{2\pi z\sqrt{-1}}$. Fix an angle $\alpha\in[0,1)$ and a strictly increasing function $f(t)$ from $\mathbb{R}_{\geq 0}$ to itself. We study a point set

[TABLE]

on a spiral curve $\{f(t)\mathop{\mathbf{e}}\nolimits(t\alpha)\ |\ t\in\mathbb{R}_{\geq 0}\}$. Clearly $X(f,\alpha)$ is not relatively dense if the angle $\alpha$ is rational, since $X(f,\alpha)$ is contained in a union of finite number of lines passing through the origin.

[TABLE]

Since $K[n]:=\{z\in X(f,\alpha)\ |\ |z|\leq f(n)\}$ has cardinality $n$ and

[TABLE]

for $f(n)>r$, we see $n\pi r^{2}\geq\pi(f(n)-r)^{2}$ which implies $2r\geq f(n)/\sqrt{n}$.

If $X(f,\alpha)$ is $s$-uniformly discrete, then $B(z,s)$ are disjoint disks for $z\in X(f,\alpha)$. From $K[n]\subset B(0,f(n))$, we obtain

[TABLE]

which leads to $n\pi s^{2}\leq\pi(f(n)+s)^{2}$, i.e., $s/2\leq f(n)/\sqrt{n}$ for $n\geq 4$. ∎

Our target is to obtain a condition that $X(f,\alpha)$ is a Delone set. It is of interest to discuss general increasing functions $f$, however, in this paper we specify $f(t)=\sqrt{t}$ and $\alpha$ is irrational and study the set

[TABLE]

From now on, we assume

[TABLE]

in light of Lemma 1.

In other words, we are interested in the sequence of points on the Fermat spiral that progresses by a constant angle $\alpha$ (see Figure 1). A real number $\alpha$ is badly approximable if there exists a positive constant $C$ so that

[TABLE]

holds for all $(p,q)\in\mathbb{Z}\times\mathbb{N}$. It is known that $\alpha$ is badly approximable if and only if partial quotients of continued fraction expansion of $\alpha$ is bounded by a positive integer $B$ (see [5, Theorem 23]). Indeed, $B$ and $C$ are roughly inverse-proportional;

[TABLE]

c.f. [3, Theorem 1.9]. We use this notation $B$ and $C$ throughout this paper. In particular, if $\alpha$ is a real quadratic irrational, then $\alpha$ is badly approximable, due to Lagrange Theorem. In this note we will prove

1. a)

$X(\sqrt{t},\alpha)$* is relatively dense,* 2. b)

$X(\sqrt{t},\alpha)$* is uniformly discrete,* 3. c)

$X(\sqrt{t},\alpha)$* is a Delone set,* 4. d)

the angle $\alpha$ is badly approximable.

By definition of a Delone set, the proof is finished when we prove the equivalence between a) and d) in §2, and then the one between b) and d) in §3. We prove them in a quantitative form.

1. a)

If $\alpha$ is badly approximable, then $X(\sqrt{t},\alpha)$ is $6\sqrt{B}$-relatively dense. 2. b)

If $X(\sqrt{t},\alpha)$ is $r$-relatively dense, then $\alpha$ is badly approximable with $B=\lfloor 96r^{2}\rfloor$. 3. c)

If $\alpha$ is badly approximable, then $X(\sqrt{t},\alpha)$ is $\sqrt{C}/2$-uniformly discrete. 4. d)

If $X(\sqrt{t},\alpha)$ is $s$-uniformly discrete, then $\alpha$ is badly approximable with $C=s^{2}/53$.

Roughly speaking, the bound $B$ corresponds to $(r,s)$-Delone set where $r$ is proportional to $\sqrt{B}$ and $s$ is inverse-proportional to $\sqrt{B}$ as in (2). As $(r,s)$-Delone set is more uniform when $r/s$ is closer to $1$, the case $B=1$, i.e., $\alpha=(\sqrt{5}-1)/2$ is the best choice. This shows an important role of the golden angle observed in many instances in phyllotaxis.

1. Three Distance Theorem

The torus $\mathbb{T}:=\mathbb{R}/\mathbb{Z}\simeq[0,1)$ is divided into subintervals by

[TABLE]

Steinhaus conjectured that there are at most three different lengths appear in this partition for any $\alpha$ and $N$. This is first proved by Sós [9, 10] and then Świerczkowski [12], Surányi [11], Halton [4] and Slater [8]. We follow a formulation in Alessandri-Berthé [1], see also [2, Chapter 2.6] and [6, Chapter 6.4, p.518]. Let $\eta_{-1}=1$, $\eta_{0}=\alpha$ and

[TABLE]

for $k\geq 0$ with a unique $a_{k+1}\in\mathbb{N}$. When $\alpha$ is irrational, the sequence $(\eta_{k})_{k=-1}^{\infty}$ is positive, strictly decreasing and given by Euclidean division process of intervals. Concatenating

[TABLE]

the continued fraction expansion of

[TABLE]

is naturally associated with this algorithm. Define $p_{-1}=1,p_{0}=0,q_{-1}=0,q_{0}=1$ and

[TABLE]

for $k\geq 0$. Then we observe that the convergent $[0,a_{1},\dots,a_{n}]$ is equal to $p_{n}/q_{n}$ and $\eta_{k}=(-1)^{k}(q_{k}\alpha-p_{k})$ by induction. We later use an important basic inequality

[TABLE]

see e.g. [7, Chapter 2] ,[5, Theorem 13], [3, Corollary 1.4]. For a positive integer $N$ there exists a unique expression:

[TABLE]

with $1\leq s\leq a_{k+1}$ and $0\leq r<q_{k}$. Three distance theorem reads

Theorem 4**.**

The torus $\mathbb{T}$ is subdivided by $\{n\alpha\pmod{1}\ |\ n\in\{0,1,\dots,N\}\}$ into intervals of at most three lengths:

• •

$N+1-q_{k}$* intervals have length $\eta_{k}$,*

• •

$r+1$* intervals have length $\eta_{k-1}-s\eta_{k}$,*

• •

$q_{k}-(r+1)$* intervals have length $\eta_{k-1}-(s-1)\eta_{k}$.*

Rotating all points by $x\mapsto x+\ell\alpha\pmod{1}$ in $\mathbb{T}$ with a fixed $\ell\in\mathbb{Z}$, the same statement is valid for $\{n\alpha\pmod{1}\ |\ n\in\{\ell,\ell+1,\dots,N+\ell\}\}$. See [9, 10, 12, 11, 4, 8] for the proof of Theorem 4. We just comment that it makes easier to understand them after we comprehend the recursive (dynamical) induction structure of irrational rotation $([0,1),x\mapsto x+\alpha)$. Inducing an irrational rotation $x\mapsto x+(-1)^{k}\eta_{k}$ on the interval $I$ of length $\eta_{k-1}$ to an interval of length $\eta_{k}$ located at the end of $I$ in the sign $(-1)^{k}$ direction, the first return map gives another irrational rotation $x\mapsto x+(-1)^{k+1}\eta_{k+1}$. As the number of points $n\alpha\pmod{1}$ increases, we inductively observe the induced irrational rotations. Three distance theorem can be easily understood as a consequence of this structure.

Corollary 5**.**

$\alpha$* is badly approximable if and only if there is a positive constant $R$, that for any $N\in\mathbb{N}$, the ratios of three distances generated by the $N+1$ points are bounded by $R$.*

Proof.

Clearly $\eta_{k-1}-(s-1)\eta_{k}$ is the sum of the other two. We obtain

[TABLE]

and

[TABLE]

On the other hand, for $n=a_{k}q_{k-1}+q_{k-2}+q_{k-1}-1=q_{k}+q_{k-1}-1$ there are exactly two lengths $\eta_{k-1}$ and $\eta_{k-2}-a_{k}\eta_{k-1}=\eta_{k}$ which appear in the partition. From $\eta_{k-1}-a_{k+1}\eta_{k}=\eta_{k+1}$ we see

[TABLE]

Summing up, the bound $B$ of partial quotients and $R$ satisfy

[TABLE]

∎

2. Diophantine condition for relative denseness

First we give a Diophantine characterization of relatively denseness. The distance of $x$ from $\mathbb{Z}$ is designated by $\|x\|=\min_{i\in\mathbb{Z}}|x-i|$. Hereafter we often use

[TABLE]

which follows from $|1-\mathop{\mathbf{e}}\nolimits(x)|=2|\sin(\pi x)|$.

Lemma 6**.**

If $X(\sqrt{t},\alpha)$ is $r$-relatively dense then for any $m\geq 16r^{2}$ and $\beta\in[0,1)$, the system of inequalities

[TABLE]

has a solution $n\in\mathbb{N}$. Conversely the system of inequalities (5) is solvable, then $X(\sqrt{t},\alpha)$ is $6r$-relatively dense.

Proof.

Assume that $X(\sqrt{t},\alpha)$ is $r$-relatively dense. Then for $z=\sqrt{m}\mathop{\mathbf{e}}\nolimits(\beta)\in\mathbb{C}$ with $m\geq 16r^{2}$, we find $n\in\mathbb{N}$ that $\sqrt{n}\mathop{\mathbf{e}}\nolimits(n\alpha)\in B(z,r)$. Then

[TABLE]

gives

[TABLE]

On the other hand,

[TABLE]

shows

[TABLE]

Conversely, take a complex number $z=\sqrt{m}\mathop{\mathbf{e}}\nolimits(\beta)$ with $m\geq 16r^{2}$ and assume the system (5) has a solution $n\in\mathbb{N}$. Then

[TABLE]

Here we used $n\geq m/4$ which follows from $m\geq 16r^{2}$. This shows for any point $z\in\mathbb{C}$ with $m\geq 16r^{2}$, the intersection $B(z,6r)\cap X(\sqrt{t},\alpha)$ is non empty. For $m<16r^{2}$, we have $|\sqrt{m}\mathop{\mathbf{e}}\nolimits(\beta)-\mathop{\mathbf{e}}\nolimits(\alpha)|<4r+1$. By (1), $X(\sqrt{t},\alpha)$ is $6r$-relatively dense. ∎

Proof of a) and b) of Theorem 3. We show that if $\alpha$ is badly approximable with the constant $R$ in Corollary 5, then the system of inequalities (5) is solvable. Take $R$ in Corollary 5 and fix $r>\max\{\sqrt{R/5},1\}$. For $m\geq 16r^{2}$, the closed interval $[m-3r\sqrt{m},m+3r\sqrt{m}]$ contains $N+1=\lfloor 3r\sqrt{m}\rfloor-\lceil-3r\sqrt{m}\rceil+1$ integers. By Theorem 4 the points

[TABLE]

gives a partition of $\mathbb{T}$ into $N+1$ intervals of at most three lengths where

[TABLE]

Every subdivided interval has length not larger than $R/(6r\sqrt{m}-1)$, because otherwise all lengths are larger than $1/(6r\sqrt{m}-1)$ which would cause overlapping. Therefore any $\beta\in\mathbb{T}$ and there exists $n$ such that

[TABLE]

which shows that (5) is solvable, recalling (1). In view of Lemma 6 and (4), to show $r$-relatively denseness of $X(\sqrt{t},\alpha)$, we need

[TABLE]

and therefore $r\geq 6\sqrt{B}$ suffices.

Let us assume that $X(\sqrt{t},\alpha)$ is $r$-relatively dense. By Lemma 6, there exists $r>0$ that for any $m\geq 16r^{2}$ and any $\beta\in\mathbb{T}$, there exists $n\in\mathbb{N}$ satisfying (5). Our goal is to find an upper bound of $a_{k}$. We may assume $a_{k+1}\geq 2$ and select $m\geq 16r^{2}$ that

[TABLE]

has cardinality $\lfloor a_{k+1}/2\rfloor q_{k}+q_{k-1}+t$ with $t\in\{1,2\}$ for 111The inequality $q_{k+1}\geq 48r^{2}$ ensures the existence of $m$ not less than $16r^{2}$. $q_{k+1}\geq 48r^{2}$. This is possible since the function

[TABLE]

from $\mathbb{R}_{\geq 0}$ to $\mathbb{N}\cup\{0\}$ satisfies

[TABLE]

By this choice of $m$, we have

[TABLE]

By Theorem 4, the torus $\mathbb{T}$ is partitioned into subintervals having three lengths $\eta_{k}$, $\eta_{k-1}-\lfloor a_{k+1}/2\rfloor\eta$ and $\eta_{k-1}-\lfloor a_{k+1}/2-1\rfloor\eta$ by

[TABLE]

From the choice of $m$, Theorem 4 and (3), a subinterval $I$ of length

[TABLE]

exists. Letting $\beta$ be the mid point of $I$, we have $\|n\alpha-\beta\|\geq a_{k+1}/(8q_{k+1})$. Summing up, we obtain

[TABLE]

by (6). This shows $a_{k+1}\leq 96r^{2}$ under the assumption $q_{k+1}\geq 48r^{2}$. Clearly $q_{k+1}<48r^{2}$ implies $a_{k+1}<48r^{2}$. Therefore we obtained $B\leq 96r^{2}$. ∎

3. Uniform discreteness of $X(\sqrt{t},\alpha)$

Finally we prove c) and d) of Theorem 3. If $X(\sqrt{t},\alpha)$ is not $(s/2)$-uniformly discrete, then we can find $n,m\in\mathbb{N}$ with $n<m$ such that

[TABLE]

Similar computation with Lemma 6 leads to

[TABLE]

Putting $k=m-n$, we have $k\|k\alpha\|<s^{2}$. Therefore $\alpha$ is not badly approximable by the constant $C=s^{2}$.

Conversely assume $\alpha$ is not badly approximable by the constant $C=s^{2}$, i.e., $\|k\alpha\|<\frac{s^{2}}{k}$ has an integer solution $k\in\mathbb{N}$. Then

[TABLE]

Putting $n=\lceil(k/s)^{2}\rceil$ and using (1), the right side is estimated by

[TABLE]

This shows that $X(\sqrt{t},\alpha)$ is not $(1/4+\pi\sqrt{5})s$-uniformly discrete.

4. Open Questions

It is of interest to generalize this result to a strictly increasing function $f$ asymptotically equal to $\sqrt{t}$, or to substitute $\mathop{\mathbf{e}}\nolimits(n\alpha)$ by a uniformly distributed sequence on $\mathbb{T}$.

Is there a higher dimensional analogy of this result ? One possibility is to find a transitive action $T$ on $S^{d-1}$, so that

[TABLE]

is a Delone set in $\mathbb{R}^{d}$.

Acknowledgments

The author came across this problem from an inspiring lecture delivered by J. F. Sadoc at RIMS Kyoto in Oct. 2018. He wishes to show his heartfelt gratitude to Y. Yamagishi at Ryukoku Univ, who organized the meeting, for advices, discussion and critical reading of the original manuscript. He is also indebted to J. Lagarias, Y. Bugeaud and A. Haynes for crucial remarks and discussion. This research is partially supported by JSPS grants (17K05159, 17H02849, BBD30028).