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

Abstract

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.