One normalized form of the <Croot-Sisask almost-periodicity theorem> is as follows. If finite sets $A,S$ in a group satisfy $|A+S|\le K|A|$, $p\ge2$, and $0<\epsilon<1$, then for every function $f$ there is $T\subseteq S$ with
$$
|T|\ge(2K)^{-O(p/\epsilon^2)}|S|
$$
such that
$$
\boxed{\|\tau_t(f*\mu_A)-f*\mu_A\|_p\le\epsilon\|f\|_p\qquad(t\in T-T).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 3 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-129/3/i/solution