Controlling Orthogonal Series with Irrational Rotations

We compute covering numbers associated to the set of partial sums of orthogonal series by means of irrational rotations. The device is a new metric inequality linking the increment's norm of partial sums to the one of ergodic averages of rotations acting on a suitable L ²-element of the torus. This allows to compute the number of balls covering the whole set of partial sums, by means of rotations.