Exercise 13, § 2

Soient C a compact subset of G and $\varepsilon >0$.

(a) Let $\alpha$ be the Haar measure of G. There exists a compact subset D of G such that $\alpha$(CD) $\leqslant (1 +\varepsilon )\alpha (D)$. (The case where $G =\mathbf{R}^n$ being immediate, one is reduced to the case where G admits a compact open subgroup H, and one may suppose that C is saturated with respect to H; one is then reduced to the case where G is discrete; one may then suppose that G is of finite type, and finally $G =\mathbf{Z}^p.$)

(b) There exists $k\in L^1(\widehat{G})$ such that $\mathscr{F}(k)\in \mathscr{K}(G)$ satisfying $\mathscr{F}(k)(x) = 1$ for $x\in C$, and $\|k\|_1\leqslant 1 +\varepsilon$. (Take $k$ of the form $\lambda f g$, where $\lambda$ is a constant and where $f, g\in L^2(\widehat{G})$ have as their Fourier transforms the characteristic functions of suitable sets, using a).)