Use <normalized Fourier analysis on a finite abelian group>. If $A\subseteq\mathbb F_2^n$ has density $\alpha$, its <large spectrum> at level $\rho>0$ is
$$
\boxed{\operatorname{Spec}_\rho(1_A)=\{t\in\mathbb F_2^n:|\widehat{1_A}(t)|\ge\rho\alpha\}.}
$$
The <Chang theorem> states that this spectrum is contained in a subspace of dimension
$$
\boxed{O(\rho^{-2}\log(\alpha^{-1})).}
$$
Equivalently, it is contained in the span of a <dissociated set> of that size.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 2 / i / Solution

For a finite abelian group $G$, use the normalized average $\mathbb E_x=|G|^{-1}\sum_x$, normalized convolution
$$
(f*g)(x)=\mathbb E_y f(x-y)g(y),
$$
and Fourier transform $\widehat f(\gamma)=\mathbb E_xf(x)\overline{\gamma(x)}$. Then Parseval's identity and the convolution identity take the forms
$$
\langle f,g\rangle=\sum_{\gamma\in\widehat G}\widehat f(\gamma)\overline{\widehat g(\gamma)},
\qquad
\widehat{f*g}(\gamma)=\widehat f(\gamma)\widehat g(\gamma).
$$


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 Incoming links: Normalized Fourier analysis on a finite abelian group