  We first focus on the finite-gap formalism for type IIB strings in AdS_3 x
S^1, which allows to encode the semiclassical spectrum of a very large family
of string solutions in a Riemann surface, the spectral curve. Then, we show
that, in the large angular momentum limit, it separates into two distinct
surfaces, allowing the derivation of an explicit expression for the spectrum,
which is correspondingly characterised by two separate branches. The latter may
be interpreted in terms of two kinds of spikes appearing on the strings:
"large" spikes, yielding an infinite contribution to the energy and angular
momentum of the string, and "small" spikes, representing finite excitations
over the background of the "large" spikes.
  On the other side of the AdS/CFT correspondence, we consider the sl(2) sector
of N=4 super Yang-Mills theory. The corresponding 1-loop spectrum, in the large
conformal spin limit, is also encoded in a spectral curve and characterised in
terms of the so-called holes. We show that, with the appropriate
identifications and with the usual extrapolation from weak to strong 't Hooft
coupling described by the cusp anomalous dimension, the large-S spectra of
gauge theory and of string theory coincide. Furthermore, we explain how "small"
and "large" holes may be identified with "small" and "large" spikes.
  Finally, we discuss several explicit spiky string solutions in AdS_3 which
display the finite-gap spectrum. We compute their spectral curves in the large
S limit, finding that they correspond to specific regions of the moduli space
of the finite-gap curves. We also explain how "large" spikes may be used in
order to extract a discrete system of degrees of freedom from string theory,
which can then be matched with the degrees of freedom of the dual gauge theory
operators, and how "small" spikes are in fact very similar to the Giant Magnons
living in R x S^2.
