Exercise 23, § 1

Let $p\geqslant 3$ be a prime number and let $k\geqslant 1$ be an integer. We denote by $G =\mathbf{F}^k_p$. We equip G with the Haar measure of total mass 1, and we identify $\widehat{G}$ and G by means of the mapping which associates to $n= (n_1, . . . , n_k)\in G$ the character

$k$

$e_n:x\mapsto$ exp$(\frac{2i\pi}{p}\sum n_jx_j)$.

$j=1$ For functions $f_1,f_2,f_3$ from G into $\mathbf{C}$, we denote by

$x\in Gh\in G$

We denote by $\|f\|$ the norm of $f$ in $L^2(G)$.

a) We have

$x\in Gx\in G$

and

$x\in G$

b) We have

and

c) Let $f: G\rightarrow \mathbf{R}$ be a function such that $\sum f(x) = 0$. There exists an affine hyperplane $H\subset G$ such that

(now prove that there exist elements $t\in G$ and $\theta \in \mathbf{R}$ such that the real part of $p^{-k}\sum_xf(x)(e_t(x)$ exp($i\theta$ ) $+ 1$) is equal to $\|\widehat{f}\|_{\infty})$.

d) Let $A\subset G$ contain no arithmetic progression of length 3, that is to say there do not exist $x\in A$ and $h\in G-\{0\}$ such that $x+h$ and $x+ 2h$ also belong to A. Show that there exists an affine hyperplane $H\subset G$ such that

Card(ACard(H)$\cap H$)$\geqslant$ Card(A)Card(G) $+\frac{1}{2}(($ Card(A)Card(G)$)^2-\frac{1}{p^k})$

(apply the preceding questions to $f=\varphi_A- |A|/p^k$, where $\varphi_A$ is the characteristic function of A).

e) Let $A\subset G$ contain no arithmetic progression of length 3. Show that Card(A) $\leqslant 3p^k/k$ (« Roth theorem for $\mathbf{F}^k_p$ »).