  We establish various forms of the following certainty principle: a set $S
\subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that
$S$ is a support of the Fourier transform of a sufficiently singular
probability measure on $\mathbb{R}^{n}$. 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.
