Given a bounded sequence of zero mean, we study the problem of its orthogonality to all continuous  observables in topological dynamical systems whose all invariant measures yield measure-theoretic systems belonging to a fixed characteristic class (a class closed under taking joinings and factors). Starting with a general Veech condition giving rise to an ergodic solution of the above problem, we will deduce that for the (characteristic) class of relative discrete spectrum, the Veech condition is equivalent to Tao's 1-Fourier uniformity condition of the original sequence.  I will show to which extent the 1-Fourier uniformity condition holds for small sets.

The talk is based on joint works with A. Kanigowski, J. Ku\l aga-Przymus, F. Richter, T. de la Rue and J. Ter\"av\"ainen.