Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$.
We recall that we denote by $(p_i)_{i \in \mathbb{N}^*}$ the sequence of prime numbers, ordered in increasing order.
Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right).$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$.
Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. We recall that $(p_i)_{i \in \mathbb{N}^*}$ denotes the sequence of prime numbers, ordered in increasing order. Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\bigcup_{i=1}^{n+1} \mathbb{N}^* r p_i = \left(\bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) \cup \left(\mathbb{N}^* r p_{n+1} \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_{n+1} p_i\right)$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the result of 17a, show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right)$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.

Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and $$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$ show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:

• $\xi \in F$,
• $\xi \cdot x \geqslant 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geqslant 0, i = 1, \ldots, k$.

Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and $$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$ show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:

• $\xi \in F$,
• $\xi \cdot x \geq 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geq 0, i = 1, \ldots, k$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geqslant 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}$$ (adopting the convention: $\inf \emptyset = +\infty$ and $\sup \emptyset = -\infty$). Show that $\alpha \geqslant \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geq 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}$$ (adopting the convention: $\inf \emptyset = +\infty$ and $\sup \emptyset = -\infty$).
Show that $\alpha \geqslant \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geqslant 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geqslant 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$

• a) Show that $p \cdot z \geqslant 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geqslant 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
• b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
• c) Show that $b \cdot \bar{q} = \alpha = \beta$.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R}), b = (b_1, \ldots, b_k) \in \mathbb{R}^k$ and $p \in \mathbb{R}^d$. Set $$\alpha := \inf\left\{p \cdot x : x \in \mathbb{R}^d, x \geq 0, Mx \leqslant b\right\}$$ and $$\beta := \sup\left\{b \cdot q : q \in \mathbb{R}^k, q \leqslant 0, M^T q \leqslant p\right\}.$$ Suppose that there exists $\bar{x} = (\bar{x}_1, \ldots, \bar{x}_d) \in \mathbb{R}^d$ such that $$\bar{x} \geq 0, M\bar{x} \leqslant b \text{ and } p \cdot \bar{x} = \alpha.$$ Denoting by $M_i$ the vector of $\mathbb{R}^d$ whose coordinates are the coefficients of the $i$-th row of $M$, set: $$I := \left\{i \in \{1, \ldots, k\} : M_i \cdot \bar{x} = b_i\right\}$$ and $$J := \left\{j \in \{1, \ldots, d\} : \bar{x}_j = 0\right\}$$
a) Show that $p \cdot z \geq 0$ for all $z \in \mathbb{R}^d$ such that $$z_j \geq 0 \text{ for all } j \in J \text{ and } M_i \cdot z \leqslant 0 \text{ for all } i \in I.$$
b) Show that there exists $\bar{q} \in \mathbb{R}^k$ such that: $$\bar{q} \leqslant 0, M^T \bar{q} \leqslant p, \bar{q} \cdot (M\bar{x} - b) = 0 \text{ and } (p - M^T \bar{q}) \cdot \bar{x} = 0.$$
c) Show that $b \cdot \bar{q} = \alpha = \beta$.

For all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, we set $$\|x\|_1 := \sum_{i=1}^d |x_i|, \quad \|x\|_\infty := \max\{|x_i|, i = 1, \ldots, d\}.$$ Show that for all $x \in \mathbb{R}^d$, we have $$\|x\|_1 = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_\infty \leqslant 1\right\}$$ and $$\|x\|_\infty = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_1 \leqslant 1\right\}.$$

For all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, we set $$\|x\|_1 := \sum_{i=1}^d |x_i|, \quad \|x\|_\infty := \max\{|x_i|, i = 1, \ldots, d\}.$$ Show that for all $x \in \mathbb{R}^d$, we have $$\|x\|_1 = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_\infty \leqslant 1\right\},$$ and $$\|x\|_\infty = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_1 \leqslant 1\right\}.$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Denote by $C$ the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Show that $C$ is non-empty, convex, closed and bounded.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$ Let $C$ be the set: $$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}$$ Show that $C$ is non-empty, convex, closed and bounded.

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \backslash \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \backslash \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|$$

Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and $$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}, \quad C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}.$$ Fix $\bar{x} \in C$. Show that there exists $q \in \operatorname{Ker}(M)^\perp \setminus \{0\}$ such that for all $i \in \{1, \ldots, d\}$, we have $$q_i \bar{x}_i = \|q\|_\infty |\bar{x}_i|.$$

Show that there exists a real $\alpha > 0$ such that
$$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$
Deduce that there exist three reals $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $t \in ] 0 , t _ { 0 } ]$ and all $\theta \in [ - \pi , \pi ]$,
$$\left| \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \right| \leq e ^ { - \beta \left( t ^ { - 3 / 2 } \theta \right) ^ { 2 } } \quad \text { or } \quad \left| \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \right| \leq e ^ { - \gamma \left( t ^ { - 3 / 2 } | \theta | \right) ^ { 2 / 3 } }$$

We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Prove that the real number $$C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$$ exists and belongs to the interval $[ 0,1 ]$.

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $R$ is $B ^ { 2 m } ( R )$ and the symplectic cylinder of radius $R'$ is $Z ^ { 2 m } ( R' ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant R'^{ 2 } \right\}$. Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.

Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.

Let $K$ be a compact set of $\mathbb{R}$. Let $k > 0$ and $B$ the set of functions from $K$ to $\mathbb{R}^d$ that are $k$-Lipschitz. Show that $B$ is equicontinuous.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. By reasoning by contradiction, show that if $A$ is relatively compact then $A$ is equicontinuous.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. We seek to show the following theorem:
Theorem 1: The following two properties are equivalent: - (P1) $A$ is relatively compact. - (P2) $A$ is equicontinuous and for all $x \in K$, the set $A(x) = \{f(x) \mid f \in A\}$ is bounded.
Show that $(P1) \Rightarrow (P2)$.