I recently needed this simple fact of harmonic analysis:

Let $G$ be a discrete abelian group and $X\subset G$ be a finite subset. Then for any $\epsilon>0$ the set $\Gamma_\epsilon=\{ \gamma\in\widehat{G} : \lvert \widehat{X}(\gamma)\rvert \geq (1-\epsilon)\lvert X\rvert\}$ is a closed subset of $\widehat{G}$ with strictly positive Haar measure; $\mu(\Gamma_\epsilon)>0$.

The point is the positivity of the Haar measure; this fails for $\epsilon=0$ or $X$ infinite, for example.

The proof is simple: observe that this set contains $\Gamma_\epsilon'-\Gamma_\epsilon'$ if $\Gamma_\epsilon'$ has the shape $$ \{ \gamma : \lvert \theta_x - \gamma(x) \rvert \leq \epsilon/2\textrm{ for all }x\in X\}$$ for any $\theta_x$, and then note that $\widehat{G}$ is covered by at most $(2\epsilon^{-1})^{\lvert X\rvert}$ many such sets.

I was surprised that I couldn't find a reference or more discussion in Rudin's `Fourier analysis on groups', and Google didn't turn up anything either (at least, nothing I could recognise as this fact). It feels like this fact is a simple corollary of some deeper fact of harmonic analysis.

Hence my questions are:

What is a reference for this fact? What is the correct generalisation to general locally compact abelian groups?

An even vaguer question: what is the iceberg of which this is a tip?