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

Table of contents
- Solution i

Solution

One normalized form of the Croot-Sisask almost-periodicity theorem is as follows. If finite sets

A, S

in a group satisfy

∣ A + S ∣ \leq K ∣ A ∣

,

p \geq 2

, and

0 < ϵ < 1

, then for every function

f

there is

T \subseteq S

with

∣ T ∣ \geq (2 K)^{- O (p / ϵ^{2})} ∣ S ∣

such that

∥ τ_{t} (f * μ_{A}) - f * μ_{A} ∥_{p} \leq ϵ ∥ f ∥_{p} (t \in T - T) .

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