l 1 - NORM OF THE FOURIER TRANSFORM ON COMPACT VECTOR SPACES

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 [3] Green and Konyagin show 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. In this paper we address, by contrast, groups with duals having rich small subgroup structure, specifically the case when G is a compact vector space over F2. The results themselves are rather technical to state but the following consequence captures their essence: If A ⊂ Fn2 is a set of density as close to 1 3 as possible then we show that ‖χA‖A(Fn 2 ) ≫ logn . We include a number of examples and conjectures which suggest that what we have shown is very far from a complete picture. 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 [4]; 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 f̂ : Ĝ → C; γ 7→ ∫