  We introduce the notion of Karamata regular operators, which is a notion of
regularity that is suitable for obtaining concrete convergence rates for common
fixed point problems. This provides a broad framework that includes, but goes
beyond, H\"olderian error bounds and H\"older regular operators. By concrete,
we mean that the rates we obtain are explicitly expressed in terms of a
function of the iteration number $k$ instead, of say, a function of the iterate
$x^k$. While it is well-known that under H\"olderian-like assumptions many
algorithms converge linearly/sublinearly (depending on the exponent), little it
is known when the underlying problem data does not satisfy H\"olderian
assumptions, which may happen if a problem involves exponentials and
logarithms. Our main innovation is the usage of the theory of regularly varying
functions which we showcase by obtaining concrete convergence rates for
quasi-cylic algorithms in non-H\"olderian settings. This includes certain rates
that are neither sublinear nor linear but sit somewhere in-between, including a
case where the rate is expressed via the Lambert W function. Finally, we
connect our discussion to o-minimal geometry and show that definable operators
in any o-minimal structure are always Karamata regular.
