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.
|Number of pages||8|
|Journal||Archiv der Mathematik|
|State||Published - Jan 2022|
Bibliographical noteFunding Information:
XS was supported by NSF Grant DMS-1802224.
© 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)