Bounds in Cohen's idempotent theorem

We show that if $G$ is a finite Abelian group and $f$ is an integer-valued map on $G$ with algebra norm at most $M$ then there is some $L < \exp(M^{4+o(1)})$,cosetsof(possiblydifferent)subgroups$W_1,…,W_L$,and$s_1,…,s_L \in {-1,1}$suchthat$f=\sum_i{s_i1_{W_i}}$.