Exercise 28, § 1

With hypotheses as in Exer. 27 a), assume in addition that there exists a $\mu$-integrable G-paving F.

a) For every $\mu$-integrable numerical function $f \geq 0$ defined on X, set $\widetilde{f}(x) = \sum_{s \in G} f(sx)$, so that $\int_F \widetilde{f}(x) d\mu(x) = \int_X f(x) d\mu(x)$ (§ 2, No. 10, Prop. 15). Let $(f_i)_{1 \leq i \leq n}$ be a family of numerical functions $\geq 0$, $\mu$-integrable on X; for $i \neq j$ set

and let $c_i = \sup_{x \in X} \widetilde{f}_i(x)$. Show that there exists at least one pair of distinct indices $(i, j)$ such that

(5) Cf. TVS II, §5, No. 2, Prop. 3; a closed convex set with nonempty interior was called a convex body (corps convexe) in the first edition of Ch. II of Esp. vect. top. (§ 3, No. 2, Def. 4).

(Bound the sum $\sum_{i \neq j} m_{ij}$ from below, using the Cauchy–Schwarz inequality.) From this, deduce that if $(A_i)_{1 \leq i \leq n}$ is a finite family of $\mu$-integrable G-fillings, and if

then there exist a pair $(i, j)$ of distinct indices and an $s \in G$ such that $\mu(A_i s \cap A_j) > 0$.

b) If $G_0$ is a subgroup of $G$ of finite index $(G : G_0) = h$, and if $(s_1, \ldots, s_h)$ is a system of representatives of the right cosets of $G_0$ in $G$, then $F_0 = \bigcup_{1 \leq i \leq h} s_i F$ is a $G_0$-paving.

c) In $X = \mathbf{R}^n$, let $A$ be a symmetric closed convex set with nonempty interior; let $u_i : (x_j) \mapsto \sum_{j=1}^n c_{ij} x_j$ be $m$ linear forms on $X$, with integer coefficients $c_{ij}$ ($m < n$). Show that for every integer $p \geq 1$ and every number $r \geq 0$ such that $\mu(A) r^n \geq 2^n p^m$, there exists a point $x \in rA \cap \mathbf{Z}^n$ distinct from 0 and such that $u_i(x) \equiv 0 \pmod{p}$ for $1 \leq i \leq m$ (apply Minkowski’s theorem (Exer. 27 c)) to the subgroup $G_0$ of $\mathbf{Z}^n$ formed by the $z \in \mathbf{Z}^n$ such that $u_i(z) \equiv 0 \pmod{p}$ for $1 \leq i \leq m$, and make use of b)). Special case that $n = 2, m = 1$ and $A$ is defined by $|x_1| \leq 1, |x_2| \leq 1$ (Thue’s theorem).