An Application of a Local Version of Chang’s Theorem

Suppose that G is a compact Abelian group. If A ⊂ G then how small can ‖χA‖A(G) be? In general there is no non-trivial lower bound. In [5] Green and Konyagin showed that if Ĝ has sparse small subgroup structure and A has density α with α(1−α) ≫ 1 then ‖χA‖A(G) does admit a non-trivial lower bound. To complement this [11] addressed the case where Ĝ has rich small subgroup structure and further claimed a result for general compact Abelian groups. In this note we prove this claim by fusing the techniques of [5] and [11] in a straightforward fashion. 1. Notation and introduction We use the Fourier transform on compact Abelian groups, the basics of which may be found in Chapter 1 of Rudin [9]; we take a moment to standardize our notation. Suppose that G is a compact Abelian group. Write Ĝ for the dual group, that is the discrete Abelian group of continuous homomorphisms γ : G → S, where S := {z ∈ C : |z| = 1}. G may be endowed with Haar measure μG normalised so that μG(G) = 1 and as a consequence we may define the Fourier transform .̂ : L(G) → l(Ĝ) which takes f ∈ L(G) to