Theorem 3.8 (Croot-Sisask almost periodicity). Assuming that:

  • G a finite abelian group

  • 𝜀>0

  • p∈[2,∞)

  • A,B⊆G are such that |A+B|≤K|A|

  • f:G→ℂ

Then there exists b∈B and a set X⊆B−b such that |X|≥2−1K−O(𝜀−2p)|B| and
∥τxf∗μA−f∗μA∥Lp(G)≤𝜀∥f∥Lp(G)∀⁡x∈X,

where τxg(y)=g(y+x) for all y∈G, and as a reminder, μA is the characteristic measure of A.