含在$\mathbb{F_p}$二次剩余中的Hilbert立方体维数的上界
Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$
摘要：
本文考虑不含任何原根的有限域$\mathbb{F_p}$中Hilbert立方体维数的界问题。我们表明，这样的希尔伯特立方体的尺寸是$O_ε(p^{1/8+ε})$的任何$ε> 0$，匹配什么可以推导出从经典的伯吉斯估计的特殊情况下，希尔伯特立方体是一个等差数列。我们还考虑了避免区间的乘法希尔伯特立方体的维数的双重问题。
英文摘要：
We consider the problem of bounding the dimension of Hilbert cubes in a finite field $\mathbb{F_p}$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
