Leinert sets and complemented ideals in Fourier algebras

We show how complemented ideals in the Fourier algebra A(G) of G arise naturally from a class of thin sets known as Leinert sets. Moreover, we present an explicit example of a closed ideal in A(F-N) where F(N)is the free group on N >= 2 generators, that is complemented in A(F-N) but it is not completely complemented. Then by establishing an appropriate extension result for restriction algebras arising from Leinert sets, we show that any almost connected group G for which every complemented ideal in A(G) is also completely complemented must be amenable.