On Scaling in Relation to Singular Spectra

Abstract.

This paper relates uniform α-Hölder continuity, or $\alpha$-dimensionality, of spectral measures in an arbitrary interval to the Fourier transform of the measure. This is used to show that scaling exponents of exponential sums obtained from time series give local upper bounds on the degree of Hölder continuity of the power spectrum of the series. The results have applications to generalized random walk, numerical detection of singular continuous spectra and to the energy growth in driven oscillators.