ISRAEL JOURNAL OF MATHEMATICS 92 (1995), 375-395 WIENER-WINTNER RETURN-TIMES ERGODIC THEOREM BY I. ASSANI* Department of Mathematics University of North Carolina at Chapel Hill, NC 27599, USA e-mail: assani@math.unc.edu AND n. LESIGNE** Ddpartement de Mathdmatiques Universitd Franqois Rabelais, Parc de Grandmont, 37200 Tours, France e-mail: LESIGNE@UNIV- Tours.FR AND D. RUDOLPH*** Department of Mathematics University of Maryland at College Park, MD 20740, USA e-mail: djr@math.umd.edu ABSTRACT We state a new ergodic theorem, combining the Wiener-Wintner theorem and Bourgain's theorem concerning the convergence of ergodic averages along return-times sequences. We consider ergodic averages of the form N-1 1 einO" f'(Sny), f(Tnx) n=O and we show that the behaviour of these averages characterizes an algebra C of functions, which contains the Kronecker algebra and has interesting properties, linked with multiple recurrence ergodic theorems. * Supported in part by NSF grant #9003929. ** Supported in part by a grant from MRT (#01-92, Pr4fecture de la R4gion Centre). *** Supported in part by NSF grant# DMS 01524638. Received December 27, 1993 375