On Hilbert cubes and primitive roots in finite fields

Ali Alsetri, Xuancheng Shao

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We consider the problem of bounding the dimension of Hilbert cubes in a finite field Fp that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is Oε(p1/8+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.

Original languageEnglish
Pages (from-to)49-56
Number of pages8
JournalArchiv der Mathematik
Issue number1
StatePublished - Jan 2022

Bibliographical note

Funding Information:
XS was supported by NSF Grant DMS-1802224.

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.


  • Character sums
  • Finite field
  • General arithmetic progression
  • Hilbert cubes
  • Multiplicative Hilbert cube
  • Primitive roots
  • Quadratic residues

ASJC Scopus subject areas

  • Mathematics (all)


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