  We study the infimum of the spectrum, or ground state energy (g.s.e.), of a
discrete Schr\"odinger operator on $\theta\mathbb{Z}^d$ parameterized by a
potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency
parameter $\theta\in (0,1)$. We relate this g.s.e. to that of a corresponding
continuous semiclassical Schr\"odinger operator on $\mathbb{R}^d$ with
parameter $\theta$, arising from the same choice of potential. We show that:
the discrete g.s.e. is at most the continuous one for continuous periodic $V$
and irrational $\theta$; the opposite inequality holds up to a factor of
$1-o(1)$ as $\theta\rightarrow 0$ for sufficiently regular smooth periodic $V$;
and the opposite inequality holds up to a constant factor for every bounded $V$
and $\theta$ with the property that discrete and continuous averages of $V$ on
fundamental domains of $\theta \mathbb{Z}^d$ are comparable. Our proofs are
elementary and rely on sampling and interpolation to map low-energy functions
for the discrete operator on $\theta \mathbb{Z}^d$ to low-energy functions for
the continuous operator on $\mathbb{R}^d$, and vice versa.
