Exercise 28, § 1

The hypotheses being those of Exer. 27 a), we suppose in addition that there exists a $\mu$-integrable G-tiling F.

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

and $c_i = \sup_{x \in X} \tilde{f}_i(x)$. Now prove that there exists at least one couple of distinct indices $(i, j)$ such that

(Minorize the sum $\sum_{i \neq j} m_{ij}$ by using the Cauchy-Schwarz inequality).

Deduce that if $(A_i)_{1 \leq i \leq n}$ is a finite family of $\mu$-integrable G-fillings, and if $\sum_{i=1}^n \mu(A_i) > \mu(F)$, there exists a couple $(i, j)$ 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 left classes mod $G_0$ in $G$, $F_0 = \bigcup_{1 \leq i \leq h} s_i F$ is a $G_0$-tiling.

c) In $X = \mathbf{R}^n$, let $A$ be a convex symmetric body; let $u_i : (x_j) \to \sum_{j=1}^n c_{ij} x_j$, be $m$ linear forms on $X$, with integer coefficients $c_{ij}$ $(m < n)$.

Now prove 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 $x \in rA$, distinct from 0 and such that $u_i(x) \equiv 0$ (mod. $p$) for $1 \leq i \leq m$ (apply Minkowski's theorem to the subgroup $G_0$ of $\mathbf{Z}^n$ formed by the $z \in \mathbf{Z}^n$ such that $u_i(z) \equiv 0$ (mod. $p$) for $1 \leq i \leq m$, and use b)). Particular case where $n = 2$, $m = 1$, and $A$ is defined by $|x_1| \leq 1,\ |x_2| \leq 1$ (Thue's theorem).