Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition. We define $$\Gamma = \left\{z \in \mathbb{R}_+^* \mid \exists (x, y) \in \Lambda,\, z = y - x\right\}, \quad \text{and} \quad r(\Lambda) = \inf \Gamma.$$ Give two examples of such sets $\Lambda$, one for which $r(\Lambda) > 0$ and another for which $r(\Lambda) = 0$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that two integer simplexes $\mathcal{S}$ and $\mathcal{S}'$ are equivalent if and only if there exist a matrix $A \in \mathrm{GL}_n(\mathbb{Z})$ and a vector $b \in \mathbb{Z}^n$ such that $\mathcal{S}' = A(\mathcal{S}) - b$.

Let $f \in \mathbb{R}^{N}$ and $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$. We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$.
Show that $F$ is differentiable and calculate its derivative $F'$. Show further that for all $\beta \in ]0, +\infty[$, there exists $p(\beta) \in \Sigma_{N}(\beta f)$ such that $F'(\beta) = -\frac{1}{\beta^{2}} H_{N}(p(\beta))$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $$\Gamma = \left\{z \in \mathbb{R}_+^* \mid \exists (x, y) \in \Lambda,\, z = y - x\right\}, \quad r(\Lambda) = \inf \Gamma.$$ We assume that $r(\Lambda) > 0$. Show that there exist $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$. Let $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$ and denote $d = b - a$. Let $k, n \in \mathbb{N}$ such that $k \leqslant n-1$. Show that $$\Lambda \cap [na + kd,\, na + (k+1)d] = \{na + kd,\, na + (k+1)d\}$$

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$. Let $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$ and denote $d = b - a$. Show that there exists $n_0 \in \mathbb{N}$ such that $n_0 a + n_0 d > (n_0 + 1)a$, then that there exists $k \in \mathbb{N}$ such that $a = kd$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$ and $d = b - a$ as defined above. Deduce that $\Lambda \subset d\mathbb{Z}$, where $d\mathbb{Z} = \{kd \mid k \in \mathbb{Z}\}$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that the volume, the number of integer points and the number of interior integer points are the same for two equivalent integer simplexes.

We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$ where $f \in \mathbb{R}^{N}$.
Study the limits of $F$ at 0 and at $+\infty$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$, $$\Lambda \cap [x, x + \eta] \neq \varnothing$$

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Suppose that for every sequence $\left(x_n\right)_{n \geqslant 0}$ with values in $\Lambda$ such that $x_n \rightarrow +\infty$, $f\left(x_n\right) \rightarrow 0$ when $n \rightarrow +\infty$. Show that $f(x) \rightarrow 0$ when $x \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Using the results of questions 13a and 13b, deduce that $h(-x) \rightarrow h(0)$ when $x \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Conclude that $h$ is a constant function.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Theorem 2 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, $\operatorname{Vol}(\mathcal{S}) \leqslant C(n,k)$.
Deduce from Theorem 2 that for every strictly positive integer $k$, there exist up to equivalence only finitely many integer simplexes in $\mathbb{R}^n$ having exactly $k$ interior points.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. The function $f$ is the unique bounded and uniformly continuous solution of equation (E), and $f'$ is bounded and uniformly continuous. Prove that the function $x \mapsto \sup_{t \geqslant x} f'(t)$ admits a finite limit when $x \rightarrow +\infty$. We denote $$c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$$

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. With $c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$, show that there exists a sequence $y_n \rightarrow +\infty$ such that $f'\left(y_n\right) \rightarrow c$ when $n \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that there exists a subsequence $\left(t_k\right)_{k \geqslant 0}$ of $\left(y_n\right)_{n \geqslant 0}$ such that the sequence of functions $\left(\xi_k\right)_{k \geqslant 0}$ defined by $$\xi_k : \mathbb{R} \rightarrow \mathbb{R}, \quad t \mapsto \xi_k(t) = f'\left(t + t_k\right)$$ converges uniformly on every segment of $\mathbb{R}$ to a function denoted $\xi$. Show that $\xi$ is constant, equal to $c$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. With $\xi$ constant equal to $c$ as shown in question 14c, conclude that $c = 0$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that $\lim_{x \rightarrow +\infty} \inf_{t \geqslant x} f'(t) = 0$. Deduce that $f'(t) \rightarrow 0$ when $t \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. Using the result that $f'(t) \rightarrow 0$ when $t \rightarrow +\infty$, show that for all $\ell \geqslant 0$, $f(t+\ell) - f(t) \rightarrow 0$ when $t \rightarrow +\infty$.

Let $\mathcal{S}$ be a simplex of $\mathbb{R}^n$ and $k$ an integer such that $\operatorname{Vol}(\mathcal{S}) > k$.
15a. Show that there exist $x \in [0,1[^n$ and $(k+1)$ elements of $\mathbb{Z}^n$ $u_0, \ldots, u_k$ such that $x \in \mathcal{S} - u_i$ for $i = 0, \ldots, k$. One may study the sets $(u + [0,1[^n) \cap \mathcal{S}$ when $u$ ranges over $\mathbb{Z}^n$; and admit — outside the CPGE curriculum — that the volume of a simplex is its Lebesgue measure, which is sub-additive.
15b. Deduce from this the existence of the $(k+1)$ points $v_0, \ldots, v_k$ that satisfy the conditions of Theorem 1.
15c. Prove Theorem 1, that is, here we assume only that $\operatorname{Vol}(\mathcal{S}) \geqslant k$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$. For all $\theta \in \mathbb{R}^{d}$, let $f(\theta) = M\theta \in \mathbb{R}^{N}$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$ and $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N}$$ where $f(\theta) = (f_{1}(\theta), \ldots, f_{N}(\theta))$. The function $L : \mathbb{R}^{d} \rightarrow \mathbb{R}$ is defined by $$L(\theta) = \ln(Z(\theta)) - q^{T} M\theta.$$
Show that $L$ is of class $\mathscr{C}^{1}$ and calculate its gradient.

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. We set $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ For all $g \in \mathscr{F}$ (the set of positive bounded functions with support in a segment of $\mathbb{R}^+$), we denote by $Lg$ the unique solution of (E) bounded with support in $\mathbb{R}^+$. A sequence $\left(t_k\right)_{k \geqslant 0}$ satisfies property $(\mathscr{P})$ if $t_k \rightarrow +\infty$ and there exists a continuous bounded function $\mu : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for every piecewise continuous $g \in \mathscr{F}$, $$Lg\left(t_k\right) \rightarrow \int_0^{+\infty} g(t)\mu(t)\,dt \quad \text{when} \quad k \rightarrow +\infty$$ Show, using question 14f, that for all $g \in \mathscr{F} \cap \mathscr{C}^1(\mathbb{R}, \mathbb{R}^+)$ and $\ell \geqslant 0$, $$\int_0^{+\infty} g(t)(\mu(t+\ell) - \mu(t))\,dt = 0$$

Under the same assumptions as question 15a, deduce that $\mu$ is constant.

Let $t_1, \ldots, t_n$ be strictly positive real numbers such that $\sum_{i=1}^n t_i = 1$ and let $N \geqslant n$ be an integer. We wish to show that there exist non-negative integers $p_1, \ldots, p_n$ and $q$ such that
i) $1 \leqslant q \leqslant N^{n-1}$,
ii) $\sum_{i=1}^n p_i = q$,
iii) $\left|qt_1 - p_1\right| \leqslant \frac{n}{N}$,
iv) for all $i = 2, \ldots, n$, $\left|qt_i - p_i\right| \leqslant \frac{1}{N}$.
16a. By considering the vectors with coordinates $\left(\{kt_2\}, \ldots, \{kt_n\}\right) \in [0,1[^{n-1}$ when $k$ ranges over $\{0, \ldots, N^{n-1}\}$, show that there exist integers $p_2, \ldots, p_n, q \geqslant 0$ satisfying conditions i) and iv).
16b. Conclude.