Symmetric cocycles and classical exponential sums

Abstract

This paper considers certain classical exponential sums as examples of cocycles with additional symmetries. Thus we simplify the proof of a result of Anderson and Pitt concerning the density of lacunary exponential partial sums P n=0 exp(2�im k x), n = 1,2,..., for fixed integer m � 2. Also, with the help of Hardy and Littlewood's approximate functional equation, but otherwise by elementary considerations, we improve a previous result of the author for certain examples of Weyl sum: if � 2 (0,1) Q has continued fraction representation (a1,a2,...) such that P n 1/an 0, then, for Lebesgue almost all x 2 (0,1), the partial sums P n=0 exp(2�i(k 2 � + 2kx)), n = 1,2,..., are dense in C. 1. Introduction. The use of cocycles to generate and study classical ex- ponential series is well established (F), (G), (Pu), (Fo), giving results which are sometimes dicult to obtain without a dynamical approach. I n this paper, exploiting an idea of symmetry that is seen most naturally from the dynam- ical point of view, we analyse two contrasting examples of complex-valued cocycle, each giving information about a corresponding exponential sum. Therst example (x2, Example 1) simplies the analysis of certain la- cunary series studied by Anderson and Pitt (AP2). The dynamics which generate such series are \hyperbolic, containing many periodic points and a rich proximal structure. The second example (x2, Example 2), taking up the greater part of the paper, is the quadratic Weyl sum. A circle extension of a rotation underlies the dynamics of this series, and such a strictly er- godic system, which is only one step removed from the rigidity of a group rotation, is opposite in most senses to the hyperbolic system of Example 1. Nevertheless, wend a useful property common to both these examples: suf- �cient symmetry; and this simple idea, described in x2, allows us to deduce strong results about the divergence of the series from comparatively weak assumptions. Before giving more detail of the results, werst describe how each of the