Bounds in Cohen's idempotent theorem (Q2310830)

For a locally compact group \(G\) let \(B(G)\) denote the Fourier-Stieltjes algebra of \(G\) and let \(A(G)\) be the Fourier algebra of \(G\). Host's theorem says that the integer-valued functions in \(B(G)\) are precisely the finite linear combinations with integer coefficients of translates of characteristic functions on open subgroups of \(G\). In particular, the idempotents in \(B(G)\) are exactly the characteristic functions on finite unions of translates of open subgroups of \(G\). In the paper under review the author investigates quantitative aspects of Host's theorem for locally compact abelian groups. Let \(\mathcal{W}(G)\) be the set of all open cosets of \(G\). The main result of this paper is: Suppose that \(M \geq 1\). Then for all finite abelian groups \(G\) and functions \(f \colon G \rightarrow \mathbb{Z}\) with \(\Vert f \Vert_{A(G)} \leq M\) there is some \(z\colon \mathcal{W}(G) \rightarrow \mathbb{Z}\) such that \[ f = \sum_{W \in \mathcal{W}(G)}z(W) 1_W \text{ and }\Vert z \Vert_{\ell_1(\mathcal{W}(G))} \leq \text{exp} \left(M^{4 + o(1)}\right). \] The author also shows that: Suppose that \(M \geq 1\). Then there is a finite abelian group \(G\) and a function \(f \colon G \rightarrow \mathbb{Z}\) with \(\Vert f \Vert_{A(G)} \leq M\) such that if \(z \colon \mathcal{W}(G) \rightarrow \mathbb{Z}\) has \[ f = \sum_{W \in \mathcal{W}(G)} z(W) 1_W, \text{ then } \Vert z \Vert_{\ell_1(\mathcal{W}(G))} = \Omega\left( \text{exp}\left(\frac{\pi^2}{4}M\right)\right).\] Now suppose \(G\) is a locally compact abelian group and let \(\mathcal{A}(G) = \{ A \subset G \colon 1_A \in B(G) \}\). Now \(\mathcal{W}(G) \subseteq \mathcal{A}(G)\). Cohen's idempotent theorem says that \(\mathcal{A}(G)\) is contained in the coset ring of \(G\). Using the idea of a \((k,s)\)-representation of a set, a quantitative version of Cohen's idempotent theorem is given. The paper is well-written and self-contained.