Norm-Controlled Inversion in Measure Algebras

We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $δ>1/2$ there is a constant $C_M(δ)$, independent of the group $G$, such that $\|μ\|_{M(G)}\le1$ and $\inf_{γ\in\widehat G}|\widehatμ(γ)|\geδ$ imply that $μ$ is invertible in $M(G)