  Let $D$ be a dictionary in a Hilbert space $H$, that is, a set of unit
elements whose linear combinations are dense in $H$. We consider the least
$m$-term deviation $\sigma_m(x)$ of an element $x\in H$: this is the distance
of $x$ from the set of all $m$-term linear combinations of elements of $D$. We
prove a dichotomy result: for any dictionary $D$, either the sequence
$\{\sigma_m(x)\}_{m=0}^{\infty}$ decreases exponentially for every $x\in H$, or
the rate of convergence $\sigma_m(x)\to 0$ can be arbitrarily slow. We seek
universal dictionaries realizing all strictly decreasing null sequences as
sequences of $m$-term deviations. All commonly used dictionaries turn out not
to be universal. In particular, the least rational deviations in Hardy space
$H^2$ do not form certain strictly monotone null sequences. There are no
universal dictionaries in finite dimensional Hilbert spaces. We construct a
universal dictionary in every infinite dimensional Hilbert space.
