The ℓ1 -norm of the Fourier transform on compact vector spaces

Abstract

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 a recent preprint, Green and Konyagin show that if G ^ 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 the complementary case of 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 ⊂ F 2 n is a set of density as close to 1/3 as possible, then we show that ‖ χ A ‖ A ( F 2 n ) ≫ log n . We include a number of examples and conjectures which suggest that what we have shown is very far from a complete picture.