  In the first part of this paper, we show that the small-ball condition,
recently introduced by Mendelson (2015), may behave poorly for important
classes of localized functions such as wavelets, piecewise polynomials or
trigonometric polynomials, in particular leading to suboptimal estimates of the
rate of convergence of ERM for the linear aggregation problem. In a second
part, we recover optimal rates of covergence for the excess risk of ERM when
the dictionary is made of trigonometric functions. Considering the bounded
case, we derive the concentration of the excess risk around a single point,
which is an information far more precise than the rate of convergence. In the
general setting of a L2 noise, we finally refine the small ball argument by
rightly selecting the directions we are looking at, in such a way that we
obtain optimal rates of aggregation for the Fourier dictionary.
