Certain isomorphisms between quotients of a group algebra

Let T be the circle group, considered as the additive group of the real numbers modulo 2. Let A = A(T) 9 the Banach algebra of functions on T which have absolutely convergent Fourier series, with the norm of / in A equal to n I f(j) \. If E is a closed subset of T, we denote by A(E) the quotient algebra A/I(E), where I(E) is the closed ideal consisting of those functions in A which vanish on E. This paper presents a procedure for constructing perfect sets E and F, which are not Helson sets, and a map : F -> E inducing an isomorphism of A{E) into A{F).