The 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 ||χ A || A ( G ) be? In general, there is no non‐trivial lower bound. In a recent preprint, Green and Konyagin show that ifG ^has sparse small subgroup structure and A has density α with α(1 − α) ≫ 1, then ||χ A || 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 2 . The results themselves are rather technical to state, but the following consequence captures their essence: if A ⊂F 2 nis 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.