  We study a one parameter family of supersymmetric marginal deformations of
${\cal N}=4$ SYM with $U(1)^3$ symmetry, known as $\beta$-deformations, to
understand their dual $AdS\times X$ geometry, where $X$ is a large classical
geometry in the $g_{YM}^2N\to \infty$ limit. We argue that we can determine
whether or not $X$ is geometric by studying the spectrum of open strings
between giant gravitons states, as represented by operators in the field
theory, as we take $N\to\infty$ in certain double scaling limits. We study the
conditions under which these open strings can give rise to a large number of
states with energy far below the string scale. The number-theoretic properties
of $\beta$ are very important. When $\exp(i\beta)$ is a root of unity, the
space $X$ is an orbifold. When $\exp(i\beta)$ close to a root of unity in a
double scaling limit sense, $X$ corresponds to a finite deformation of the
orbifold. Finally, if $\beta$ is irrational, sporadic light states can be
present.
