Arithmetric structure and lacunary Fourier series

bounded function on E can be extended to be a Fourier-Stieltjes transform. It is known that every Sidon set is a A set, but the converse is unknown [4, p. 128]. As a corollary to Theorem 2 we find that A sets are also Sidon sets in groups F where each element has the same prime order. NOTATION. Suppose s and N are positive integers, 1 _r oo, b1, * * , bk E F have finite order, bk+1, b , bc F have infinite order and bo E F has arbitrary order. Set Ar(N, s, b) = n b + bo ZIn, Jr < N where n*b=nlbl+*+nsbs. Denote by f the Fourier transform of fe L'(G), by (x, y) the action of the character y on x E G, and by IFl the cardinality of a finite set F. THEOREM 1. Let bl, * * *, bk E r have finite orders ,l1, P*k, and let bk1, * * *, bs F F have infinite order. Let Ec F be a A(p) set with B a Received by the editors September 24, 1971 and, in revised form, December 8, 1971. AMS 1970 subject classifications. Primary 42A44, 43A70.