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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-129/2/i/solution