Lipschitz conditions and lacunarity

We consider 2π-periodic functions on the limits and give simple and complete characterizations, in terms of Fourier coefficients, of functions which belong to various Lipschitz classes and whose Fourier series are lacunary. Such characterisations seem to be missing from the literature, though there are various wellknown partial characterisations valid for functions with arbitrary spectra; cf. the remarks following Theorem 1. The results given below form complements to and sharpenings of some of the standard results valid for the special case of lacunary series.