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We consider the problem of bounding the dimension of Hilbert cubes in afinite field $\mathbb{F_p}$ that does not contain any primitive roots. We showthat 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 Burgessestimate in the special case when the Hilbert cube is an arithmeticprogression. We also consider the dual problem of bounding the dimension ofmultiplicative Hilbert cubes avoiding an interval.

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$