  We consider the problem of bounding the dimension of Hilbert cubes in a
finite field $\mathbb{F_p}$ that does not contain any primitive roots. We show
that the dimension of such Hilbert cubes is $O_{\epsilon}(p^{1/8+\epsilon})$
for any $\epsilon > 0$, matching what can be deduced from the classical Burgess
estimate in the special case when the Hilbert cube is an arithmetic
progression. We also consider the dual problem of bounding the dimension of
multiplicative Hilbert cubes avoiding an interval.
