A solution to the pyjama problem

Abstract

The “pyjama stripe” is the subset of $\mathbb {R}^2$ consisting of a vertical strip of width $2 \varepsilon$ around every integer $x$-coordinate. The “pyjama problem” asks whether finitely many rotations of the pyjama stripe around the origin can cover the plane. The purpose of this paper is to answer this question in the affirmative, for all positive $\varepsilon$. The problem is reduced to a statement closely related to Furstenberg’s $\times 2,\, \times 3$ Theorem from topological dynamics, and is proved by analogy with that result.

Appendix: Proofs of auxiliary results

Here we provide proofs of some of the less standard and more technical results of Sect. 6.

1.1 Torsion points and periodic points

Proof of Proposition 6.4

Consider (i). Unwrapping the definitions, we have that $n\, x = 0$ for $n \in \mathbb {Z}$ if and only if there is some $r \in A$ such that $n\, z = \imath _\mathbb {C}(r)$, $n\, a = \imath _{\mathbb {Q}_5}(r)$, $n\, b = \imath _{\mathbb {Q}_{13}}(r)$. Setting $q = r / n$ gives (a $\Rightarrow$ b) and choosing $n \in \mathbb {Z}$ such that $n\, q \in A$ gives (b $\Rightarrow$ a).

For (c $\Rightarrow$ a), observe that by pigeonhole there exist $\phi \ne \psi \in \Theta$ such that $\phi (x) = \psi (x)$, whence $(\phi - \psi ) (x) = 0$ and so, multiplying by $a \in \mathbb {Z}[i]$ such that $a (\phi - \psi ) \in \mathbb {Z}$, we deduce $x$ is torsion.

Finally, for (a, b $\Rightarrow$ c) we note that $n\, x = 0 \Rightarrow n \theta (x) = 0$ for $\theta \in \Theta$, so it suffices to check that, for given $n$, the set $\{ x \in \widehat{A} \,:\, n\, x = 0 \}$ is finite. This is actually a group, and by (b) is isomorphic under $\imath ^\Delta$ to the quotient $\{ q \in \mathbb {Q}(i) \,:\, n\, q \in A \} / A$, which in turn is isomorphic (under $q \mapsto n q$) to $A / n A$. That this last group is finite is a standard fact.

We now turn to (ii). For (a $\Rightarrow$ b), we write $\phi = (\theta _{5}\, \theta _{13})^m$ and suppose $\phi (x) = x$. That means there exists $r \in A$ such that $\imath _\mathbb {C}(\phi - 1) z = \imath _\mathbb {C}(r)$, $\imath _{\mathbb {Q}_5}(\phi - 1) a = \imath _{\mathbb {Q}_5}(r)$, $\imath _{\mathbb {Q}_{13}}(\phi - 1) b = \imath _{\mathbb {Q}_{13}}(r)$, so taking $q = r / (\phi - 1)$ it suffices to check that $|q|_{\mathcal {P}_5},\, |q|_{\mathcal {P}_{13}} \le 1$. Note $|r|_{\mathcal {P}_5},\, |r|_{\mathcal {P}_{13}} \le 1$ (since $r \in A$) and $|\phi |_{\mathcal {P}_5},\, |\phi |_{\mathcal {P}_{13}} < 1$ since $m > 0$, so $|\phi - 1|_{\mathcal {P}_5},\, |\phi - 1|_{\mathcal {P}_{13}} = 1$.

For (b $\Rightarrow$ a), we observe as above that the orbit $\Theta (x)$ is contained in

$$\begin{aligned} \{ x \in \widehat{A} \,:\, n\, x = 0 \} \cong A / n\, A \end{aligned}$$

for some $n \in \mathbb {Z}$, which crucially can now be taken to be coprime to $65$. It follows that the corresponding multiplicative action of $\Theta$ on the ring $A / n\, A$ is now invertible, i.e. the image of $\Theta$ lies in $(A / n\, A)^\times$. Hence we can choose $m$ to be $|(A / n\, A)^\times |$. $\square$

1.2 Unions of complex balls

Proof of Proposition 6.6

We prove (i). Let $x_1, \dots , x_k \in Y$ be such that $\bigcup _i \overline{B}_1^\mathbb {C}(x_i)$ covers $Y$. It suffices to show each $x_i$ has the form $x_i = y_i + \jmath _\mathbb {C}(z_i)$ where $y_i$ is torsion and $z_i \in \mathbb {C}$.

By pigeonhole, there exist distinct $\theta _1, \theta _2 \in \Theta ^{(m)}$ such that $\theta _1(x_i)$ and $\theta _2(x_i)$ lie in the same ball $\overline{B}_1^\mathbb {C}(x_j)$. Hence $\theta _1(x_i) - \theta _2(x_i) \in \jmath _{\mathbb {C}}(\mathbb {C})$. Choosing $r \in \mathbb {Z}[i]$ such that $r (\theta _1 - \theta _2) = n \in \mathbb {Z}$ we obtain $n\, x_i \in \jmath _{\mathbb {C}}(\mathbb {C})$ as required.

Now consider (ii). $\Rightarrow$ follows trivially from (i): $\overline{\Theta ^{(m)}(x)}$ is contained in a finite union of complex balls, is closed and invariant, and hence is contained in a finite union of complex balls with torsion centers, which a fortiori implies $x$ itself has the specified form.

For $\Leftarrow$, we invoke Proposition 6.4 (i)(a $\Rightarrow$ c) and note the trivial fact that the orbit closure of a point $\jmath _\mathbb {C}(w)$, $w \in \mathbb {C}$ is contained in (in fact, is equal to) $\jmath _\mathbb {C}\left( \{ z \in \mathbb {C}\,:\, |z| = |w| \}\right)$, which is certainly contained in some $\overline{B}_R^\mathbb {C}(0)$. $\square$

1.3 Non-Archimedean limits

Proof of Proposition 6.10

By the hypothesis on $Y$, we can pick a sequence $x_i \in Y$ such that $x_i - x_j \notin \overline{B}_{1}^\mathbb {C}(0)$ for all $i \ne j$. Since $Y$ is compact metric, passing to some subsequence we have $x_i \rightarrow x$ for some $x \in Y$.

It would suffice to show that $x_i - x \notin \overline{B}_{1/2}^\mathbb {C}(0)$ for all $i$, as then $x_i \rightarrow x$ and $x_i - x \rightarrow 0$ would be non-Archimedean limits in $Y$ and $Y - Y$ respectively.

In fact this holds for all $i$ with at most one exception: if $x_i - x \in \overline{B}_{1/2}^\mathbb {C}(0)$ and $x_j - x \in \overline{B}_{1/2}^\mathbb {C}(0)$ for $i \ne j$ then $x_i - x_j \in \overline{B}_1^\mathbb {C}(0)$, contradicting the choice of the $x_i$. Deleting the exceptional index (if there is one) gives the result. $\square$

1.4 Density of $\mathbb {C}$, $\mathbb {Q}_5$ and $\mathbb {Q}_{13}$ in $\widehat{A}$

We need a standard fact for the proof of Proposition 6.11.

Proposition 8.1

Any finite index multiplicative subgroup of $\mathbb {Z}_p^\times$ contains a subgroup $(1 + p^n \mathbb {Z}_p, \times )$ for some $n$.

Proof

There is an isomorphism $(1 + p \mathbb {Z}_p, \times ) \longleftrightarrow (\mathbb {Z}_p, +)$ given by the $p$-adic $\log$ and $\exp$ maps (for $p > 2$); see e.g. [10, Chapter II, Section 3, Proposition 8].

So if $J \le \mathbb {Z}_p^\times$ has finite index then $J \cap (1 + p \mathbb {Z}_p)$ has finite index in $1 + p \mathbb {Z}_p$, which corresponds to a finite index subgroup of $(\mathbb {Z}_p, +)$ under the isomorphism; those are all $(p^n \mathbb {Z}_p, +)$ for some $n$, which correspond to $(1 + p^{n+1} \mathbb {Z}_p, \times )$ under the isomorphism the other way. $\square$

Corollary 8.2

Let $J$ be a finite index subgroup of $\mathbb {Q}_p^\times$. There exists some $\eta > 0$ depending only on $J$, such that if $x, y \in \mathbb {Q}_p^\times$ and $|x - y|_p / |x|_p \le \eta$ then $x$ and $y$ are in the same coset of $J$.

Proof

We require $y / x \in J$. By Proposition 8.1 applied to the finite index subgroup $J_0 = J \cap \mathbb {Z}_p^\times \le \mathbb {Z}_p^\times$, there is some $\eta > 0$ such that if $a \in \mathbb {Z}_p^\times$ and $|1 - a|_p \le \eta$ then $a \in J_0$. So—provided $\eta \le 1$—we have $|1 - a|_p \le \eta \Rightarrow a \in J_0$ for all $a \in \mathbb {Q}_p^\times$. But $|1 - y / x|_p = |x - y|_p / |x|_p$ and the result follows. $\square$

We isolate one further result from the proof of Proposition 6.11.

Lemma 8.3

Let $J,\, H$ be as in the statement of Proposition 6.11. For any $\mu , \nu > 0$ we can find an $r \in A$ such that $|r|_\mathbb {C},\, |r|_{\overline{\mathcal {P}_{13}}} \le \mu$, $|r|_{\overline{\mathcal {P}_5}} \ge \nu$ and $\imath _{\mathbb {Q}_5}(r) \in H$. The same holds with $5$ and $13$ exchanged.

Proof

Let $x_1, \dots , x_k \in A$ be coset representatives for $J$; i.e. $\bigcup _i x_i\, J = \mathbb {Q}_5^\times$. (This is possible by Corollary 8.2 and the fact that $A$ is dense in $\mathbb {Q}_p^\times$.) Let $C = \max \left\{ |x_i|_\mathbb {C},\, |x_i|_{\overline{\mathcal {P}_{13}}},\, 1 / |x_i|_{\overline{\mathcal {P}_5}} \right\}$. Pick any $y \in A$ such that $|y|_\mathbb {C},\, |y|_{\overline{\mathcal {P}_{13}}} < 1$ and $|y|_{\overline{\mathcal {P}_5}} > 1$ (say, $y = \overline{\mathcal {P}_{13}} / \overline{\mathcal {P}_5}^{10}$). Then choose $n$ large enough that $|y^n|_\mathbb {C},\, |y^n|_{\overline{\mathcal {P}_{13}}} \le \mu / C$ and $|y^n|_{\overline{\mathcal {P}_5}} \ge C\, \nu$. Now take $i$ such that $x_i\, y^n \in H$, and observe $r = x_i\, y^n$ has the desired properties. $\square$

Proof of Proposition 6.11

Let $x = \jmath (z, a, b) \in \widehat{A}$, with $(z, a, b)$ in the fundamental domain (say). We take $\delta > 0$ arbitrary, and choose $q \in A$ such that $|\imath _\mathbb {C}(q) - z|_\mathbb {C},\, |\imath _{\mathbb {Q}_{13}}(q) - b|_{13} \le \delta / 4$ (by Proposition 6.12).

If $a - \imath _{\mathbb {Q}_5}(q) \in H$ we are happy, as then

$$\begin{aligned} d_{\widehat{A}}((z, a, b),\, (0,\, a - \imath _{\mathbb {Q}_5}(q),\, 0)) = d_{\widehat{A}}((z, a, b),\, (\imath _\mathbb {C}(q),\, a,\, \imath _{\mathbb {Q}_{13}}(q)) \le \delta / 2. \end{aligned}$$

If not, the above results allow us to perturb $q$ to some $q - r$ so that this holds (i.e. $a - \imath _{\mathbb {Q}_5}(q - r) \in H$), without affecting the other properties of $q$ too much.

Specifically, take $\eta > 0$ as in Corollary 8.2, and $r$ as in Lemma 8.3 with parameters $\mu = \delta / 4$ and $\nu = 10 |a - \imath _{\mathbb {Q}_5}(q)|_5 / \eta$. We claim $a - \imath _{\mathbb {Q}_5}(q - r) \in H$. Indeed, as $\imath _{\mathbb {Q}_5}(r) \in H$ by construction and $|a - \imath _{\mathbb {Q}_5}(q)|_5 / |\imath _{\mathbb {Q}_5}(r)|_5 \le \eta / 10$, this follows by Corollary 8.2. Finally,

$$\begin{aligned}&d_{\widehat{A}}((z, a, b),\, (0, a - \imath _{\mathbb {Q}_5}(q - r), 0))\nonumber \\&\quad = d_{\widehat{A}}((z, a, b),\, (\imath _\mathbb {C}(q - r),\, a,\, \imath _{\mathbb {Q}_{13}}(q - r)) \\&\quad \le |z - \imath _\mathbb {C}(q)|_\mathbb {C}+ |r|_\mathbb {C}+ |\imath _{\mathbb {Q}_{13}}(q) - b|_{\overline{\mathcal {P}_{13}}} + |r|_{\overline{\mathcal {P}_{13}}} \le \delta \end{aligned}$$

which, as $\delta$ was arbitrary, completes the proof. $\square$

1.5 Proof of Proposition 7.4

Proof of Proposition 7.4

Using [10, Chapter II, Section 3, Proposition 7], we write $u = \xi \, v$ where $\xi$ is a $(p-1)^{\mathrm {st}}$ root of unity and $v \in 1 + p \mathbb {Z}_p$. By assumption $v \ne 1$. It suffices to show $\overline{\{v^{(p-1)r} :\, r \ge 0\}}$ is a finite index subgroup of $(1 + p \mathbb {Z}_p, \times )$. Using the $p$-adic $\log$/$\exp$ isomorphism ([10, Chapter II, Section 3, Proposition 8]) this reduces to showing that for $x \in \mathbb {Z}_p$ non-zero, the closure $\overline{\{(p-1) r\, x \,:\, r \ge 0\}}$ is a finite index subgroup of $(\mathbb {Z}_p, +)$; and in fact it is precisely $p^{v_p(x)} \mathbb {Z}_p$. $\square$