Bounds in Cohen’s Idempotent Theorem

Abstract

AbstractSuppose that G is a finite Abelian group and write $${\mathcal {W}}(G)$$W(G) for the set of cosets of subgroups of G. We show that if $$f:G \rightarrow {\mathbb {Z}}$$f:G→Z satisfies the estimate $$\Vert f\Vert _{A(G)} \le M$$‖f‖A(G)≤M with respect to the Fourier algebra norm, then there is some $$z:{\mathcal {W}}(G) \rightarrow {\mathbb {Z}}$$z:W(G)→Z such that $$\begin{aligned} f=\sum _{W \in {\mathcal {W}}(G)}{z(W)1_W}\quad \text { and }\quad \Vert z\Vert _{\ell _1({\mathcal {W}}(G))} =\exp (M^{4+o(1)}). \end{aligned}$$f=∑W∈W(G)z(W)1Wand‖z‖ℓ1(W(G))=exp(M4+o(1)).