A class of lacunary trigonometric series

It is shown that there exists a sequence of natural numbers {n_k} which does not belong to the class B₂ and which cannot be decomposed into a finite number of lacunary sequences such that: a) if the series converges on a set of positive measure, then the series consisting of the squares of the coefficients converges; b) for each set E of positive measure we can remove from the system a finite number of terms with the result that what is left is a Bessel system in L²(E); and c) if the series converges to zero on a set of positive measure, then each coefficient is zero.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 781–788, December, 1973.In conclusion the author wishes to thank V. F. Emel'yanov for posing the problem and for helping to solve it.