On finite configurations in the spectra of singular measures

Abstract

We establish various forms of the following certainty principle: a set S Ă R n contains a given finite linear pattern, provided that S is a support of the Fourier transform of a sufficiently singular probability measure on R n . As its main corollary, we provide new dimensional estimates for PDE-and Fourier-constrained vector measures. Those results, in certain cases of restrictions given by homogeneous operators, improve known bounds related to the notion of the k-wave cone.