7 Rational Versions

Definition 44

✓

We define the two functions $\sin _{\mathbb {Q}}, \cos _{\mathbb {Q}}: \operatorname{\mathbb {R}}\to \operatorname{\mathbb {R}}$ by:

\begin{align*} \sin _{\mathbb {Q}}(x) & :=x-\frac{x^3}{3}+\frac{x^5}{5!}\mp \dots +\frac{x^{25}}{25!},\\ \cos _{\mathbb {Q}}(x) & :=1-\frac{x^2}{2}+\frac{x^4}{4!}\mp \dots +\frac{x^{24}}{24!}. \end{align*}

Further, by replacing $\sin ,\cos $ with $\sin _{\mathbb {Q}},\cos _{\mathbb {Q}}$ we define the functions

\[ R_{\operatorname{\mathbb {Q}}}(\alpha ), R'_{\operatorname{\mathbb {Q}}}(\alpha ), X_{\operatorname{\mathbb {Q}}}(\theta , \varphi ), M_{\operatorname{\mathbb {Q}}}(\theta , \varphi ), M_{\operatorname{\mathbb {Q}}}^{\theta }(\theta ,\varphi ),M_{\operatorname{\mathbb {Q}}}^{\varphi }(\theta ,\varphi ). \]

Lemma 45

✓

\[ |\sin _{\mathbb {Q}}(x)-\sin (x)|\leq \frac{|x|^{27}}{27!} \quad \text{and} \quad |\cos _{\mathbb {Q}}(x)-\cos (x)|\leq \frac{|x|^{26}}{26!}. \]

Proof ▶

Appeal to Taylor series bounds, using the fact that all absolute values of higher derivatives of sine and cosine never exceed 1.

Lemma 46

✓

For every $x\in [-4,4]$ it holds that

\[ |\sin _{\mathbb {Q}}(x)-\sin (x)| \leq \frac{\kappa }{7} \quad \text{and} \quad |\cos _{\mathbb {Q}}(x)-\cos (x)|\leq \frac{\kappa }{7}. \]

Proof ▶

Straightforward numerical calculation from Lemma 45.

Lemma 47

✓

Let $A = (a_{i,j})_{1 \leq i \leq m,\ 1 \leq j \leq n} \in \operatorname{\mathbb {R}}^{m \times n}$ and $\delta {\gt}0$. Assume that $|a_{i,j}| \leq \delta $. Then it holds that $\| A\| \leq \delta \sqrt{mn}.$

Proof ▶

For any $v\in \operatorname{\mathbb {R}}^n$ we have

\begin{align*} \| Av\| ^2 & =\sum _{i=1}^m \left(\sum _{j=1}^na_{i,j}v_j\right)^2 \leq \sum _{i=1}^m\left(\sum _{j=1}^n \delta |v_j|\right)^2 = \delta ^2 m\left(\sum _{j=1}^n |v_j|\right)^2 \leq \delta ^2 m n \| v\| ^2 \end{align*}

using the Cauchy-Schwarz inequality. Dividing by $\| v\| $ and taking the square root proves the claim.

Lemma 48

✓

Let $A(x,y)$ be an $m\times n$ matrix with $1 \leq m,n\leq 3$ such that every entry is of the form $a_1(x)\cdot a_2(y)$ where $a_i(z)\in \{ 0,1,-1,\pm \sin (z),\pm \cos (z)\} .$ Define $A_{\mathbb {Q}}(x,y)$ by replacing $\sin $ with $\sin _{\mathbb {Q}}$ and $\cos $ with $\cos _{\mathbb {Q}}$. Then for every $x,y\in [-4,4]$ it holds that $\| A(x,y)-A_{\mathbb {Q}}(x,y)\| \leq \kappa $.

Proof ▶

We’ve replaced the assumption $a_i(z)\in \{ 0,1,-1,\pm \sin (z),\pm \cos (z)\} $ in [ SY25 ] ’s Lemma 40 with $a_i(z)\in [-1,1]$.

By assumption, for fixed $x,y$ every entry of $A(x,y)-A_{\mathbb {Q}}(x,y)$ is of the form $a b - \widetilde{a}\widetilde{b}$ for some $a,b\in [-1,1]$ and $|a-\widetilde{a}|,|b-\widetilde{b}|\leq \kappa /7$ by lemma 46. This implies that

\begin{align*} |ab-\widetilde{a}\widetilde{b}|& \leq |a b-a\widetilde{b}|+|a \widetilde{b}-\widetilde{a}\widetilde{b}| =|a|\cdot |b-\widetilde{b}|+|\widetilde{b}|\cdot |a-\widetilde{a}| \leq 1\cdot \kappa /7+(1+\kappa /7) \cdot \kappa /7 {\lt}\kappa /3. \end{align*}

So we can apply lemma 47 and obtain that $\| A(x,y)-A_{\operatorname{\mathbb {Q}}}(x,y)\| {\lt}\kappa /3\cdot \sqrt{3\cdot 3}=\kappa $.

Corollary 49

Let $\alpha ,\theta ,\varphi \in [-4,4]$. Then it holds that

\begin{align*} \| R(\alpha )-R_{\operatorname{\mathbb {Q}}}(\alpha )\| , \| R’(\alpha )-R_{\operatorname{\mathbb {Q}}}’(\alpha )\| ,\| X(\theta ,\varphi )-X_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| , \| M(\theta , \varphi )-M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| , \\ \| M^\theta (\theta ,\varphi )-M_{\operatorname{\mathbb {Q}}}^\theta (\theta ,\varphi )\| , \| M^\varphi (\theta ,\varphi ) - M_{\operatorname{\mathbb {Q}}}^\varphi (\theta ,\varphi )\| \leq \kappa . \end{align*}

Moreover,

\[ \| R_{\operatorname{\mathbb {Q}}}(\alpha )\| , \| R'_{\operatorname{\mathbb {Q}}}(\alpha )\| , \| X_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| , \| M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\| , \| M_{\operatorname{\mathbb {Q}}}^{\theta }(\theta ,\varphi )\| , \| M_{\operatorname{\mathbb {Q}}}^{\varphi }(\theta ,\varphi )\| \leq 1+\kappa \]

Proof ▶

The first statement is a direct application of lemma 48 and the second statement follows immediately after using lemma 12 and the triangle inequality.

Lemma 50

✓

For $1 \leq i \leq n$ let $(A_i,B_i)$ be pairs of real matrices, such that for each $i$ the dimensions of $A_i$ and $B_i$ are equal. Assume moreover that the products $A_1\cdots A_n$ and $B_1 \cdots B_n$ are well defined. Finally, assume that $\| A_i-B_i\| \leq \kappa $ and let $\delta _i\geq \max (\| A_i\| ,\| B_i\| ,1)$. Then it holds that $\| A_1\cdots A_n-B_1\cdots B_n\| \leq n\kappa \cdot \delta _1\cdots \delta _n$.

Proof ▶

See [ SY25 ] , Lemma 42.

Lemma 51

Let $\alpha , \theta , \varphi \in [-4,4]$, $P\in \operatorname{\mathbb {R}}^3$ with $\| P\| \leq 1$ and let $\widetilde{P}$ be a $\kappa $-rational approximation of $P$. Set $M = M(\theta , \varphi )$ and $M_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$, $M^\theta = M^\theta (\theta , \varphi )$, $M^\theta _{\operatorname{\mathbb {Q}}} = M^\theta _{\operatorname{\mathbb {Q}}}(\theta , \varphi )$, $M^\varphi = M^\varphi (\theta , \varphi )$, $M^\varphi _{\operatorname{\mathbb {Q}}} = M^\varphi _{\operatorname{\mathbb {Q}}}(\theta , \varphi )$ as well as $R = R(\alpha )$, $R_{\operatorname{\mathbb {Q}}} = R_{\operatorname{\mathbb {Q}}}(\alpha )$, $R' = R'(\alpha )$, $R'_{\operatorname{\mathbb {Q}}} = R'_{\operatorname{\mathbb {Q}}}(\alpha )$. Finally let $w \in \operatorname{\mathbb {R}}^2$ with $\| w\| = 1$. Then:

\begin{align} | \langle M P, w\rangle - \langle M_{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 3\kappa , \label{eq:boundskappa1} \\ | \langle M^\theta P, w\rangle - \langle M^\theta _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 3\kappa ,\\ | \langle M^\varphi P, w\rangle - \langle M^\varphi _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 3\kappa ,\\ | \langle R M P, w\rangle - \langle R_{\operatorname{\mathbb {Q}}} M_{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa ,\label{eq:boundskappa4} \\ | \langle R’ M P, w\rangle - \langle R’_{\operatorname{\mathbb {Q}}} M_{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa ,\\ | \langle R M^\theta P, w\rangle - \langle R_{\operatorname{\mathbb {Q}}} M^\theta _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa ,\\ | \langle R M^\varphi P, w\rangle - \langle R_{\operatorname{\mathbb {Q}}} M^\varphi _{\operatorname{\mathbb {Q}}} \widetilde{P}, w\rangle | & \leq 4\kappa . \end{align}

Proof ▶

See [ SY25 ] , Lemma 44.

Theorem 52 Rational Global Theorem

Let $\operatorname{\mathbf{P}}$ be a pointsymmetric convex polyhedron with radius $\rho =1$ and $\widetilde{\operatorname{\mathbf{P}}}$ a $\kappa $-rational approximation. Let $\widetilde{S} \in \widetilde{\operatorname{\mathbf{P}}}$. Further let $\varepsilon {\gt}0$ and $\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {Q}}\cap [-4,4]$. Let $w\in \operatorname{\mathbb {Q}}^2$ be a unit vector. Denote $\overline{M_1}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1, \overline{\varphi }_1)$, $ \overline{M_2}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2, \overline{\varphi }_2)$ as well as $\overline{M_1}^{\theta } :=M_{\operatorname{\mathbb {Q}}}^\theta (\overline{\theta }_1, \overline{\varphi }_1)$, $\overline{M_1}^{\varphi } :=M_{\operatorname{\mathbb {Q}}}^\varphi (\overline{\theta }_1, \overline{\varphi }_1)$ and analogously for $\overline{M_2}^{\theta }, \overline{M_2}^{\varphi }$. Finally set

\begin{align*} G^{\operatorname{\mathbb {Q}}}& :=\langle R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}\widetilde{S},w \rangle - \varepsilon \cdot \big(|\langle R_{\operatorname{\mathbb {Q}}}’(\overline{\alpha }) \overline{M_1}\widetilde{S},w \rangle |+|\langle R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}^\theta \widetilde{S},w \rangle |+|\langle R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}^\varphi \widetilde{S},w \rangle |\big) \\ & \hspace{11cm}- 9\varepsilon ^2/2 - 4\kappa ( 1 + 3 \varepsilon ),\\ H^{\operatorname{\mathbb {Q}}}_P & :=\langle \overline{M_2}P,w \rangle + \varepsilon \cdot \big(|\langle \overline{M_2}^\theta P,w \rangle |+|\langle \overline{M_2}^\varphi P,w \rangle |\big) + 2\varepsilon ^2 + 3\kappa ( 1+2\varepsilon ). \end{align*}

If $G^{\operatorname{\mathbb {Q}}}{\gt}\max _{P\in \widetilde{\operatorname{\mathbf{P}}}} H^{\operatorname{\mathbb {Q}}}_P$ then there does not exist a solution to Rupert’s condition to $\operatorname{\mathbf{P}}$ with

\[ (\theta _1,\varphi _1,\theta _2,\varphi _2,\alpha ) \in [\overline{\theta }_1\pm \varepsilon ,\overline{\varphi }_1\pm \varepsilon ,\overline{\theta }_2\pm \varepsilon ,\overline{\varphi }_2\pm \varepsilon ,\overline{\alpha }\pm \varepsilon ]. \]

Proof ▶

Definition 53

Let $\theta , \varphi \in \operatorname{\mathbb {Q}}\cap [-4,4]$ and $M_{\operatorname{\mathbb {Q}}} :=M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$. Three points $\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3 \in \operatorname{\mathbb {Q}}^3$ with $\| \widetilde{P}_1\| , \| \widetilde{P}_2\| , \| \widetilde{P}_3\| \leq 1+\kappa $ are called $\varepsilon $-$\kappa $-spanning for $(\theta , \varphi )$ if it holds that:

\begin{align*} \langle R(\pi /2) M_{\operatorname{\mathbb {Q}}} \widetilde{P}_1,M_{\operatorname{\mathbb {Q}}} \widetilde{P}_{2}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ) + 6\kappa ,\\ \langle R(\pi /2) M_{\operatorname{\mathbb {Q}}} \widetilde{P}_2,M_{\operatorname{\mathbb {Q}}} \widetilde{P}_{3}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ) + 6\kappa ,\\ \langle R(\pi /2) M_{\operatorname{\mathbb {Q}}} \widetilde{P}_3,M_{\operatorname{\mathbb {Q}}} \widetilde{P}_{1}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ) + 6\kappa . \end{align*}

Lemma 54

Let $P_1, P_2, P_3 \in \operatorname{\mathbb {R}}^3$ with $\| P_i\| \leq 1$ and $\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3 \in \operatorname{\mathbb {Q}}^3$ be their $\kappa $-rational approximations. Assume that $\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3$ are $\varepsilon $-$\kappa $-spanning for some $\theta , \varphi \in \operatorname{\mathbb {Q}}\cap [-4,4]$, then $P_1, P_2, P_3$ are $\varepsilon $-spanning for $\theta , \varphi $.

Proof ▶

See [ SY25 ] , Lemma 46.

Lemma 55

Let $P,Q \in \operatorname{\mathbb {R}}^3$ with $\| P\| ,\| Q\| \leq 1$ and $\widetilde{P},\widetilde{Q}$ some respective $\kappa $-rational approximations. Moreover, let $\alpha , \theta , \varphi \in \operatorname{\mathbb {R}}\in [-4,4]$ and set $X = X(\theta , \varphi )$, $X_{\operatorname{\mathbb {Q}}} = X_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$ as well as $M = M(\theta , \varphi )$, $M_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )$. Then

\begin{align} |\langle X, P \rangle - \langle X_{\operatorname{\mathbb {Q}}}, \widetilde{P} \rangle | & \leq 3 \kappa , \label{eq:boundskappa3.1}\\ |\langle MP, MQ \rangle - \langle M_{\operatorname{\mathbb {Q}}} \widetilde{P}, M_{\operatorname{\mathbb {Q}}}\widetilde{Q} \rangle | & \leq 5 \kappa , \label{eq:boundskappa3.3}\\ |\| M Q \| - \| M_{\operatorname{\mathbb {Q}}}\widetilde{Q} \| | & \leq 3 \kappa .\label{eq:boundskappa3.2} \end{align}

Proof ▶

See [ SY25 ] , Lemma 49.

Corollary 56

In the setting of lemma 55 let additionally $\overline{\theta }, \overline{\varphi }\in \operatorname{\mathbb {R}}\cap [-4,4]$ and set $\overline{M} = M(\overline{\theta }, \overline{\varphi })$, $\overline{M}_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\overline{\theta }, \overline{\varphi })$. Then

\[ |\| R(\alpha ) M P - \overline{M} Q\| - \| R_{\operatorname{\mathbb {Q}}}(\alpha ) M_{\operatorname{\mathbb {Q}}} \widetilde{P} - \overline{M}_{\operatorname{\mathbb {Q}}} \widetilde{Q}\| | \leq 6 \kappa .\label{eq:boundskappa3.4} \]

Proof ▶

See [ SY25 ] , Corollary 50.

Corollary 57

In the setting of lemma 55, let $\sqrt[+]{x}$ be an upper-$\operatorname{\mathbb {Q}}$-square-root function and set $\| x\| _{+} :=\sqrt[+]{\| x\| ^2}$. Set

\[ A = \frac{\langle M P, M(P-Q)\rangle - 2 \varepsilon \| P-Q\| \cdot (\sqrt{2}+\varepsilon )}{ \big(\| M P\| +\sqrt{2} \varepsilon \big) \cdot \big(\| M(P-Q)\| +2 \sqrt{2} \varepsilon \big)} \]

as well as

\[ A_{\operatorname{\mathbb {Q}}} = \frac{\langle M_{\operatorname{\mathbb {Q}}} \widetilde{P}, M_{\operatorname{\mathbb {Q}}} (\widetilde{P}-\widetilde{Q})\rangle - 10\kappa - 2 \varepsilon ( \| \widetilde{P}-\widetilde{Q}\| _{+} + 2 \kappa ) \cdot (\sqrt{2}+\varepsilon )}{ \big(\| M_{\operatorname{\mathbb {Q}}} \widetilde{P}\| _{+}+\sqrt{2} \varepsilon + 3\kappa \big) \cdot \big(\| M_{\operatorname{\mathbb {Q}}}(\widetilde{P}-\widetilde{Q})\| _{+}+2 \sqrt{2} \varepsilon + 6\kappa \big)}. \]

Then it holds that $A \geq A_{\operatorname{\mathbb {Q}}}$.

Proof ▶

See [ SY25 ] , Corollary 51.

Theorem 58 Rational Local Theorem

Let $\operatorname{\mathbf{P}}$ be a polyhedron with radius $\rho =1$ and $\widetilde{P}_i$ be a $\kappa $-rational approximation of $P_i \in \operatorname{\mathbf{P}}$. Set $\widetilde{\operatorname{\mathbf{P}}} = \{ \widetilde{P}_i \text{ for } P_i \in \operatorname{\mathbf{P}}\} $. Let $P_1, P_2, P_3, Q_1, Q_2, Q_3 \in \operatorname{\mathbf{P}}$ be not necessarily distinct and assume that $P_1, P_2, P_3$ and $Q_1, Q_2, Q_3$ are congruent. Let $\varepsilon {\gt}0$ and $\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {Q}}\cap [-4,4]$. Set $\overline{X_1}:=X_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1,\overline{\varphi }_1), \overline{X_2}:=X_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2,\overline{\varphi }_2)$ as well as $\overline{M_1}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1,\overline{\varphi }_1), \overline{M_2}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2,\overline{\varphi }_2)$. Assume that there exist $\sigma _P, \sigma _Q \in \{ 0,1\} $ such that

\[ (-1)^{\sigma _P} \langle \overline{X_1},\widetilde{P}_i\rangle {\gt}\sqrt{2}\varepsilon + 3\kappa \quad \text{and} \quad (-1)^{\sigma _Q} \langle \overline{X_2}, \widetilde{Q}_i\rangle {\gt}\sqrt{2}\varepsilon + 3\kappa , \tag {A$^{\operatorname{\mathbb {Q}}}_\varepsilon $} \]

for all $i=1,2,3$. Moreover, assume that $\widetilde{P}_1,\widetilde{P}_2,\widetilde{P}_3$ are $\varepsilon $-$\kappa $-spanning for $(\overline{\theta }_1,\overline{\varphi }_1)$ and that $\widetilde{Q}_1,\widetilde{Q}_2,\widetilde{Q}_3$ are $\varepsilon $-$\kappa $-spanning for $(\overline{\theta }_2,\overline{\varphi }_2)$. Let $\sqrt[+]{x}$ and $\sqrt[-]{x}$ be upper- and lower-$\operatorname{\mathbb {Q}}$-square-root functions, then set $\| Z\| _{+} :=\sqrt[+]{\| Z\| ^2}$ and $\| Z\| _{-} :=\sqrt[-]{\| Z\| ^2}$ for $Z \in \operatorname{\mathbb {Q}}^n$. Finally, assume that for all $i = 1,2,3$ and any $\widetilde{Q}_j \in \widetilde{\operatorname{\mathbf{P}}} \setminus \widetilde{Q}_i$ it holds that

\[ \frac{\langle \overline{M_2}\widetilde{Q}_i,\overline{M_2}(\widetilde{Q}_i-\widetilde{Q}_j)\rangle - 10\kappa - 2 \varepsilon ( \| \widetilde{Q}_i-\widetilde{Q}_j\| _{+} + 2 \kappa ) \cdot (\sqrt{2}+\varepsilon )}{ \big(\| \overline{M_2}\widetilde{Q}_i\| _{+}+\sqrt{2} \varepsilon + 3\kappa \big) \cdot \big(\| \overline{M_2}(\widetilde{Q}_i-\widetilde{Q}_j)\| _{+}+2 \sqrt{2} \varepsilon + 6\kappa \big)} {\gt} \frac{\sqrt{5} \varepsilon + \delta }{r}, \tag {B$^{\operatorname{\mathbb {Q}}}_\varepsilon $} \]

for some $r {\gt}0$ such that $\min _{i=1,2,3}\| \overline{M_2}\widetilde{Q}_i \| _{-} {\gt} r + \sqrt{2} \varepsilon + 3\kappa $ and for some $\delta \in \operatorname{\mathbb {R}}$ with

\[ \delta = \max _{i=1,2,3}\left\| R_{\operatorname{\mathbb {Q}}}(\overline{\alpha }) \overline{M_1}\widetilde{P}_i-\overline{M_2}\widetilde{Q}_i\right\| _{+}/2 + 3\kappa . \]

Then there exists no solution to Rupert’s problem $R(\alpha ) M(\theta _1,\varphi _1)\operatorname{\mathbf{P}}\subset M(\theta _2,\varphi _2)\operatorname{\mathbf{P}}^\circ $ with

\[ (\theta _1, \varphi _1, \theta _2, \varphi _2, \alpha ) \in [\overline{\theta }_1\pm \varepsilon ,\overline{\varphi }_1\pm \varepsilon ,\overline{\theta }_2\pm \varepsilon ,\overline{\varphi }_2\pm \varepsilon ,\overline{\alpha }\pm \varepsilon ] \subseteq \operatorname{\mathbb {R}}^5. \]

Proof ▶