Herbert S. Zuckerman

Singular measures with absolutely continuous convolution squares

Introduction . A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivasev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such that for every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has p th power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

SOME THEOREMS ON LACUNARY FOURIER SERIES, WITH EXTENSIONS TO COMPACT GROUPS

1. Introduction. 1.1. The theory of lacunary Fourier series, developed by Banach, Kolmogorov, Sidon, Zygmund, and others, deals largely with extraordinary properties that such series possess. The Fourier series for a bounded function converges absolutely if the function has a lacunary Fourier series; an integrable function is square integrable if its Fourier series is lacunary. Somewhat different theorems show that very general functions on lacunary sets can be represented by Fourier or Fourier-Stieltjes transforms. Many of these theorems are given in [7, Chapter VI and Chapter IX].

Singular measures with absolutely continuous convolution squares.(Corrigendum)

Mr Denis Lichtman has kindly pointed out to us that an added hypothesis is needed in Theorem (4·1). The revised theorem follows.

On a theorem of P. J. Cohen and H. Davenport

where the integral is the Haar integral on G, R is some positive constant not depending on G, and N is sufficiently large. For the case in which G is the circle group, H. Davenport [2] has improved (0) by replacing the exponent | by f and the constant R by §. Cohens and Davenports arguments can in all likelihood be combined to yield (0) with exponent \ for an arbitrary G such that X is torsionfree. In this note we apply Davenports ideas to prove (0) with exponent ï not only for the case of torsion-free X but also for the case in which the torsion subgroup of X is an arbitrary finite Abelian group. By using care in our estimates we find some fairly large possible Rs, and we also work out some numerical cases. If X has infinite torsion subgroup, we show that no inequality like (0) can possibly hold.