Perturbing PLA

Abstract

We proved earlier that every measurable function on the circle, after a uniformly small perturbation, can be written as a power series (i.e. a series of exponentials with positive frequencies), which converges almost everywhere. Here we show that this result is basically sharp: the perturbation cannot be made smooth or even Hölder. We discuss also a similar problem for perturbations with lacunary spectrum.

21 pages