A new class of rank one transformations with singular spectrum ∗

A new class of rank one transformations with singular spectrum * Abstract. We introduce a new tool to study the spectral type of rank one transformations using the method of central limit theorem for trigonometric sums. We get some new applications. 1. Introduction The purpose of this paper is to bring a new tool in the study of the spectral type of rank one transformations. Rank one transformations have simple spectrum and in [O] D.S. Ornstein, using a random procedure, produced a family of mixing rank one transformations. It follows that the Ornstein's class of transformations may possibly contain a candidate for Banach's well-known problem whether there exists a dynamical system (Ω, A, µ, T) with simple Lebesgue spectrum. But, in 1993, J. Bourgain in [B] proved that almost surely Ornstein's transformations have singular spectrum. Subsequently, using the same method, I. Klemes [Kl] and I. Klemes & K. Reinhold [K-R] obtain that mixing staircase transformations of Adams [A] and Adams & Friedman [AF] have singular spectrum. They conjecture that all rank one transformations have singular spectrum. Here we shall exhibit a new class of rank one transformations with singular spectrum. Our assumption include some new class of Ornstein transformations and a class of Creutz-Silva rank one transformations [C-S]. Our proof is based on techniques introduced by J. Bourgain [B] in the context of rank one transformations and developed by Klemes [Kl], Klemes-Rienhold [K-R], Dooley-Eigen [E-D], together with some ideas from the proof of the central limit theorem for trignometric