On Hilbert cubes and primitive roots in finite fields

被引：0
作者：
Ali Alsetri
Xuancheng Shao
机构：
[1] University of Kentucky,Department of Mathematics
来源：
Archiv der Mathematik | 2022年 / 118卷
关键词：
Hilbert cubes; Primitive roots; Quadratic residues; Finite field; General arithmetic progression; Character sums; Multiplicative Hilbert cube; Primary 11N25; Secondary 11B25; 11L40;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
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.
引用
收藏
页码：49 / 56
页数：7
相关论文
共 23 条
  • [1] Bourgain J(2009)Multilinear exponential sums in prime fields under optimal entropy condition on the sources Geom. Funct. Anal. 18 1477-1502
  • [2] Burgess DA(1962)On character sums and primitive roots Proc. Lond. Math. Soc. 3 179-192
  • [3] Chang MC(2008)On a question of Davenport and Lewis and new character sum bounds in finite fields Duke Math. J. 145 409-442
  • [4] Dietmann R(2012)Hilbert cubes in progression-free sets and in the set of squares Israel J. Math. 192 59-66
  • [5] Elsholtz C(2015)Hilbert cubes in arithmetic sets Rev. Mat. Iberoam. 31 1477-1498
  • [6] Dietmann R(2013)On gaps between primitive roots in the Hamming metric Q. J. Math. 64 1043-1055
  • [7] Elsholtz C(2013)On gaps between quadratic non-residues in the Euclidean and Hamming metrics Indag. Math. (N. S.) 24 930-938
  • [8] Dietmann R(2017)Prescribing the binary digits of squarefree numbers and quadratic residues Trans. Amer. Math. Soc. 369 8369-8388
  • [9] Elsholtz C(1999)On Hilbert cubes in certain sets Ramanujan J. 3 303-314
  • [10] Shparlinski IE(2015)Character sums over unions of intervals Forum Math. 27 3017-3026
123