  In frustrated spinel antiferromagnets, dilution with non-magnetic ions can be
a powerful strategy for probing unconventional spin states or uncovering
interesting phenomena. Here, we present X-ray, neutron scattering and
thermodynamic studies of the effects of magnetic dilution of the
tetragonally-distorted A-site spinel antiferromagnet, CuRh$_2$O$_4$, with
non-magnetic Zn$^{2+}$ ions. Our data confirm the helical spin order recently
identified at low-temperatures in this material, and further demonstrate a
systematic suppression of the associated N\'eel temperature with increasing
site dilution towards a continuous transition with critical doping of $x_{spin}
\sim 0.44$. Interestingly, this critical doping is demonstrably distinct from a
second structural critical point at $x_{JT} \sim 0.6$, which is consistent with
the suppression of orbital order on the A-site through a classical percolative
mechanism. This anomalously low value for $x_{spin}$ is confirmed via multiple
measurements, and is inconsistent with predictions of classical percolation
theory, suggesting that the spin transition in this material is driven by an
enhancement of pre-existing spin fluctuations with weak dilution.
