  Consider an operator that takes the Fourier transform of a discrete measure
supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a
compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest
singular value when $\Omega$ is either a ball or cube of radius $m$, and under
different types of geometric assumptions on $\mathcal{X}$. We first show that
if distances between points in $\mathcal{X}$ are lower bounded by a $\delta$
that is allowed to be arbitrarily small, then the smallest singular value is at
least $Cm^{d/2} (m\delta)^{\lambda-1}$, where $\lambda$ is the maximum number
of elements in $\mathcal{X}$ contained within any ball or cube of an explicitly
given radius. This estimate communicates a localization effect of the Fourier
transform. While it is sharp, the smallest singular value behaves better than
expected for many $\mathcal{X}$, including when we dilate a generic set by
parameter $\delta$. We next show that if there is a $\eta$ such that, for each
$x\in\mathcal{X}$, the set $\mathcal{X}\setminus\{x\}$ locally consists of at
most $r$ hyperplanes whose distances to $x$ are at least $\eta$, then the
smallest singular value is at least $C m^{d/2} (m\eta)^r$. For dilations of a
generic set by $\delta$, the lower bound becomes $C m^{d/2} (m\delta)^{\lceil
(\lambda-1)/d\rceil }$. The appearance of a $1/d$ factor in the exponent
indicates that compared to worst case scenarios, the condition number of
nonharmonic Fourier transforms is better than expected for typical sets and
improve with higher dimensionality.
