On Hilbert cubes and primitive roots in finite fields View Full Text


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Article Info

DATE

2021-10-30

AUTHORS

Ali Alsetri, Xuancheng Shao

ABSTRACT

We consider the problem of bounding the dimension of Hilbert cubes in a finite field Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_p$$\end{document} that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is Oε(p1/8+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_{\varepsilon }(p^{1/8+\varepsilon })$$\end{document} for any ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}, 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. More... »

PAGES

49-56

References to SciGraph publications

  • 2013-04-05. Burgess’s Bounds for Character Sums in NUMBER THEORY AND RELATED FIELDS
  • 2012-05-12. Hilbert cubes in progression-free sets and in the set of squares in ISRAEL JOURNAL OF MATHEMATICS
  • 1999-09. On Hilbert Cubes in Certain Sets in THE RAMANUJAN JOURNAL
  • 2009-01-08. Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources in GEOMETRIC AND FUNCTIONAL ANALYSIS
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    http://scigraph.springernature.com/pub.10.1007/s00013-021-01672-3

    DOI

    http://dx.doi.org/10.1007/s00013-021-01672-3

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