Bounds in Cohen\u2019s Idempotent Theorem

Abstract

Suppose that G is a finite Abelian group and write {mathcal {W}}(G) for the set of cosets of subgroups of G. We show that if f:G rightarrow {mathbb {Z}} satisfies the estimate Vert fVert _{A(G)} le M with respect to the Fourier algebra norm, then there is some z:{mathcal {W}}(G) rightarrow {mathbb {Z}} such that f=∑W∈W(G)z(W)1Wand‖z‖ℓ1(W(G))=exp(M4+o(1)).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f=\\sum _{W \\in {\\mathcal {W}}(G)}{z(W)1_W}\\quad \\text { and }\\quad \\Vert z\\Vert _{\\ell _1({\\mathcal {W}}(G))} =\\exp (M^{4+o(1)}). \\end{aligned}$$\\end{document}