Ergodic Theory and Dynamical Systems

Research Article

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

Mustafa Akcoglua1, Alexandra Bellowa2, Roger L. Jonesa3, Viktor Loserta4, Karin Reinhold-Larssona5 and Máté Wierdla6§

a1 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada

a2 Department of Mathematics, Northwestern University, Evanston, IL 60201, USA

a3 Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, IL 60614, USA

a4 Institute of Mathematics, University of Vienna, A-1090 Vienna, Austria

a5 Department of Mathematics, SUNY at Albany, Albany NY 12222, USA

a6 Department of Mathematics, Northwestern University, Evanston, IL 60201, USA

Abstract

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

(Received March 08 1994)

(Revised June 10 1994)

Footnotes

† Partially supported by an NSERC grant.

‡ Partially supported by NSF grant DMS 9108746.

§ Partially supported by an NSF grant.