Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of \mathbbF_p

2021/08/06 by Ali Alsetri, Alsetri, Ali, Xuancheng Shao +1
Mathematics · #Advanced Topology and Set Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2108.03286

openalex publication_date 2021/08/06 · openalex created_date 2021/08/16 · openalex updated_date 2026/07/28

Abstract

We consider the problem of bounding the dimension of Hilbert cubes in a finite field \mathbbF_p that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is O_ε(p^1/8+ε) for any ε> 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.

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