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

TL;DR

This paper establishes an upper bound on the dimension of Hilbert cubes within finite fields that exclude primitive roots, aligning with classical estimates, and explores related dual problems.

Contribution

It provides a new upper bound of $O_{ ext{ε}}(p^{1/8+ε})$ for the dimension of such Hilbert cubes, extending classical Burgess estimates to this context.

Findings

Bound on Hilbert cube dimension is $O_{ ext{ε}}(p^{1/8+ε})$.

Matches classical Burgess estimate in special cases.

Explores dual problem of multiplicative Hilbert cubes avoiding intervals.

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.