  In this paper we quantify the notion of antisymmetry of the Fourier transform
of certain vector valued measures. The introduced scale is related to the
condition appearing in Uchiyama's theorem and is used to give a lower bound for
the rectifiable dimension of those measures. Moreover, we obtain an estimate of
the lower Hausdorff dimension assuming certain more restrictive version of the
$2$-wave cone condition. Results of our considerations can be viewed as an
uncertainty-type principle in the following way: it is impossible to
simultaneously localize a (bundle) measure and a direction of its Fourier
transform on small sets. The investigated class is modeled on the example of
gradients of BV functions. The article contains also a theorem concerning
regularity: we prove that elements of considered class vanish on 1-purely
unrectifiable sets. Our results can be applied to studying the properties of
PDE-constrainted measures.
