We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. Conclude that for all $g$ of $\mathscr{F}$ piecewise continuous, $$\sum_{k=0}^{+\infty} \mathbb{E}\left(g\left(x - S_k\right)\right) \rightarrow \frac{1}{\mathbb{E}(X)} \int_{-\infty}^{+\infty} g(t)\,dt \quad \text{when} \quad x \rightarrow +\infty$$

Let $\mathcal{S}$ be an integer simplex of $\mathbb{R}^n$ with vertices $0, s_1, \ldots, s_n$ having exactly $k$ interior integer points and let $x = \sum_{i=1}^n t_i s_i$ be an interior integer point of $\mathcal{S}$.
18a. Show that $\sum_{i=1}^n t_i \leqslant 1 - \alpha(k,n)$. (One may reason by contradiction and construct then $k+1$ distinct integer points interior to $\mathcal{S}$.)
18b. Show that $\frac{\alpha(k,n)}{1-\alpha(k,n)} x \in (\mathcal{S} - x)$.
18c. Deduce that $a(\mathcal{S} - x) \geqslant \frac{\alpha(k,n)}{1-\alpha(k,n)}$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ where $\widetilde{M} = (M \mid \mathbf{1})$.
We are interested in this question in the number of points at which the function $L$ attains its minimum.
(a) Show that if $\theta$ and $\theta'$ are two distinct points of $\mathbb{R}^{N}$ such that $L$ has a critical point at $\theta$, then the derivative of $t \rightarrow L(t\theta + (1-t)\theta')$ is strictly increasing on $[0,1]$ and vanishes at $t = 1$.
(b) Deduce that there is at most one critical point for $L$ and conclude on the number of points at which $L$ attains its minimum.

Let $\ell > 0$ be fixed. Determine the behaviour of $\mathbb{E}(N(x, x+\ell))$ when $x \rightarrow +\infty$. Interpret the result. Is this result true if there exists $d > 0$ such that $\mathbb{P}(X \in d\mathbb{Z}) = 1$?

Conclude the proof of Theorem 2, which states: For every strictly positive integer $k$, there exists a strictly positive constant $C(n,k)$ such that for every integer simplex $\mathcal{S}$ in $\mathbb{R}^n$ having exactly $k$ interior integer points, $\operatorname{Vol}(\mathcal{S}) \leqslant C(n,k)$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ and that the function $L$ has a global minimum attained at $\theta_{*}$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
(a) Show that $H_{N}(p(\theta_{*})) \geqslant H_{N}(q)$ and then that $H_{N}(p(\theta_{*}))$ is the maximum value of $H_{N}$ on $\Sigma_{N}(\bar{g}, g)$.
(b) Show that $p(\theta_{*})$ is the unique point of $\Sigma_{N}(\bar{g}, g)$ at which $H_{N}$ attains its maximum.

a) For any matrix $M \in M _ { n } ( \mathbb { C } )$ and any real number $C > 0$, show the equivalence $$\| M \| \leqslant C \Longleftrightarrow \forall x \in \mathbb { C } ^ { n } : \| M x \| _ { 1 } \leqslant C \| x \| _ { 1 } .$$ b) Show that the map $M \longmapsto \| M \|$ is a norm on $M _ { n } ( \mathbb { C } )$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. This space is equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying the following hypotheses: (H1) $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. (H2) $M^2 = \operatorname{Id}_E$. (H3) $\forall (v,w) \in E^2, (M(v) \mid w) = (v \mid M(w))$. (H4) $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$. We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
For any vector $v \in E$, we set $$v^+ = v + M(v), \quad v^- = v - M(v)$$ (a) Show that $\forall v \in E, v^+ \in F^+$ and $v^- \in F^-$.
(b) Show that $E = F^+ \oplus^\perp F^-$.
(c) Show that $\forall v \in F^+, T(v) \in F^-$ and that $\forall v \in F^-, T(v) \in F^+$.
Deduce that $F^+$ and $F^-$ are stable under $T^2$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Show that for all $k \in \{0, 1, \ldots, 2m\}$, $\operatorname{Im}(T^{k+1}) \subset \operatorname{Im}(T^k)$ and $\operatorname{Im}(T^{k+1}) \neq \operatorname{Im}(T^k)$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Deduce that for all $k \in \{0, \ldots, 2m+1\}$, we have $$\operatorname{dim}(\operatorname{Im}(T^k)) = 2m+1-k, \quad \operatorname{dim}(\operatorname{ker}(T^k)) = k$$

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Deduce also that $\operatorname{Im}(T^k) = \operatorname{ker}(T^{2m+1-k})$ for $0 \leq k \leq 2m+1$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Let $k \in \{1, 2, \ldots, 2m+1\}$ and $z \in \operatorname{Im}(T^k)^\perp \cap \operatorname{Im}(T^{k-1})$ such that $z \neq 0_E$. After justifying the existence of such a vector $z$, show that $T^{2m+1-k}(z) \neq 0_E$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Show that for any real number $\alpha$, the endomorphism $\operatorname{Id}_E + \alpha T^2$ is bijective and that $$\left(\operatorname{Id}_E + \alpha T^2\right)^{-1} = \sum_{k=0}^{m} (-1)^k \alpha^k T^{2k}$$ where $\left(\operatorname{Id}_E + \alpha T^2\right)^{-1}$ denotes the inverse endomorphism of $\operatorname{Id}_E + \alpha T^2$.

Show that for all $f \in \mathscr { C } _ { b } ^ { 2 }$, $f$ admits an entropy relative to $\mu$ and that $$\operatorname { Ent } _ { \mu } ( f ) \leqslant \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } \mu ( x ) d x$$ You may consider the family of functions defined by $f _ { \delta } = \delta + f ^ { 2 }$ for $\delta > 0$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Show that $\int \left( 1 + | x | + x ^ { 2 } \right) m ( x ) d x < + \infty$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f \in \mathscr { C } _ { b } ^ { 1 }$. We wish to show that $f$ admits a variance relative to $m$ and that $$\operatorname { Var } _ { m } ( f ) \leqslant \frac { C } { 2 } \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{2}$$
10a. Show that $fm$ and $f ^ { 2 } m$ are integrable, and that it suffices to show (2) in the case where we additionally have $\int f ( x ) m ( x ) d x = 0$ and $\int f ( x ) ^ { 2 } m ( x ) d x = 1$.
10b. Under the hypotheses of the previous question, show (2). You may apply (1) to the family of functions $f _ { \varepsilon } = 1 + \varepsilon f$ for $\varepsilon > 0$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f$ be a function in $\mathscr { C } _ { b } ^ { 1 }$, such that for all $x \in \mathbb { R }$, we have $\left| f ^ { \prime } ( x ) \right| \leqslant 1$. We denote, for $\lambda \in \mathbb { R }$, $$H ( \lambda ) = \int e ^ { \lambda f ( x ) } m ( x ) d x$$ We admit that $H$ is of class $\mathscr { C } ^ { 1 }$ and that we obtain an expression of $H ^ { \prime } ( \lambda )$ by differentiating under the integral sign in the usual manner.
11a. Show that for all $\lambda \in \mathbb { R }$, $$\lambda H ^ { \prime } ( \lambda ) - H ( \lambda ) \ln H ( \lambda ) \leqslant \frac { C \lambda ^ { 2 } } { 4 } H ( \lambda )$$
11b. Deduce that for $\lambda \geqslant 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ You may study the function $\lambda \mapsto \frac { 1 } { \lambda } \ln H ( \lambda )$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that inequality (3) $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right)$$ applies to all $f \in \mathscr{C}_b^1$ with $|f'(x)| \leq 1$. Show that inequality (3) applies to the function defined by $f ( x ) = x$. You may use the sequence of functions defined by $f _ { n } ( x ) = n \arctan \left( \frac { x } { n } \right)$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that for $\lambda \geq 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ applies (in particular to $f(x) = x$).
13a. Let $M = \int x m ( x ) d x$ and $a \geqslant M$. Show that $$\int _ { a } ^ { + \infty } m ( x ) d x \leqslant \exp \left( - \frac { ( a - M ) ^ { 2 } } { C } \right)$$
13b. Conclude that for all $\alpha < \frac { 1 } { C }$, the function $x \mapsto e ^ { \alpha x ^ { 2 } } m ( x )$ is integrable on $\mathbb { R }$.

Let $p , q , r : \mathbb { R } \rightarrow \mathbb { R } _ { * } ^ { + }$ be three continuous functions, with strictly positive values and integrable on $\mathbb { R }$.
14a. Show that there exists a function $u : ] 0,1 [ \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 1 }$ bijective such that $$\forall t \in ] 0,1 [ , \quad u ^ { \prime } ( t ) p ( u ( t ) ) = \int p ( x ) d x$$ Similarly, there exists an analogous function $v : ] 0,1 [ \rightarrow \mathbb { R }$ for $q$.
14b. We assume that $$\forall x , y \in \mathbb { R } , \quad p ( x ) q ( y ) \leqslant \left( r \left( \frac { x + y } { 2 } \right) \right) ^ { 2 } . \tag{4}$$ Show that $$\left( \int p ( x ) d x \right) \left( \int q ( x ) d x \right) \leqslant \left( \int r ( x ) d x \right) ^ { 2 } \tag{5}$$ You may use, after having justified its validity, the change of variable defined by $x = \frac { u ( t ) + v ( t ) } { 2 }$ in the right-hand side of inequality (5).

We recall that $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$ is a measure, and that for $A \in \operatorname{Int}$, $\mu(A) = \int \mathbb{1}_A(x) \mu(x) dx$. We denote $d(x, A) = \inf\{|x - y| : y \in A\}$. Let $A \subset \mathbb { R }$.
15a. Show that for all $x , y \in \mathbb { R }$, we have $$\exp \left( \frac { 1 } { 2 } d ( x , A ) ^ { 2 } - x ^ { 2 } \right) \mathbb { 1 } _ { A } ( y ) \exp \left( - y ^ { 2 } \right) \leqslant \exp \left( - \frac { ( x + y ) ^ { 2 } } { 2 } \right)$$
15b. We assume that $A \in \operatorname { Int }$ and that $\mu ( A ) > 0$. Deduce that $$\int \exp \left( \frac { 1 } { 2 } d ( x , A ) ^ { 2 } \right) \mu ( x ) d x \leqslant \frac { 1 } { \mu ( A ) }$$

We recall that $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$ is a measure, and that for $A \in \operatorname{Int}$, $\mu(A) = \int \mathbb{1}_A(x) \mu(x) dx$. Let $A \in \operatorname{Int}$. For $t \geqslant 0$, we define the set $A _ { t } = \{ x \in \mathbb { R } : d ( x , A ) \leqslant t \}$.
16a. Show that $A _ { t } \in \operatorname { Int }$ for all $t \geqslant 0$.
16b. We further assume that $\mu ( A ) > 0$. Show that for all $t \geqslant 0$, we have $$1 - \mu \left( A _ { t } \right) \leqslant \frac { e ^ { - t ^ { 2 } / 2 } } { \mu ( A ) }$$

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Let $\mathcal { S }$ be the set defined in question 13: $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$ Show that the set of elements of $\mathcal { S }$ whose characteristic polynomial $P$ has roots of multiplicity at most 2 in $\mathbb { C }$ is dense in $\mathcal { S }$.
Hint: You may use $r \left( P ^ { \prime } \right)$ where the map $r$ is defined in question 11.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. In light of the results of questions 24 and 25, what can we conclude regarding the relationship between propositions $\left( \mathcal { F } _ { 1 } \right)$ and $\left( \mathcal { F } _ { 2 } \right)$?

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The subspace $G = \mathbb{R}_{2m-1}^0[X]$ (polynomials of degree at most $2m-1$ vanishing at $\pm 1$).
Let $(P_1, \ldots, P_{2m-2})$ be any basis of $G$. We consider the two square matrices $A = [a_{i,j}]_{1 \leq i,j \leq 2m-2}$ and $B = [b_{i,j}]_{1 \leq i,j \leq 2m-2}$ defined by $$a_{i,j} = (P_i \mid P_j), \quad b_{i,j} = (P_i' \mid P_j')$$ Determine the ratio $$\frac{\operatorname{det}(A)}{\operatorname{det}(B)}$$ as a function of $m$.