Bounds in Cohen’s Idempotent Theorem

Suppose that G is a finite Abelian group and write 𝒲(G) for the set of cosets of subgroups of G . We show that if f:G →ℤ satisfies the estimate ‖ f‖ _A(G)≤ M with respect to the Fourier algebra norm, then there is some z:𝒲(G) →ℤ such that f=∑ _W ∈𝒲(G)z(W)1_W and ‖ z‖ _ℓ _1(𝒲(G)) =exp (M^4+o(1)).