有限域 $\mathbf{F}_{q^r}$ 中避开仿射超平面的本原元，对于 $q=4$ 和 $q=5$

$\mathbf{Bad}(s,t)$ is hyperplane absolute winning

J. An (2013) proved that for any $s,t \geq 0$ such that $s + t = 1$, $\mathbf{Bad}(s,t)$ is $(34\sqrt 2)^{-1}$-winning for Schmidt's game. We show that using the main lemma from An's paper one can derive a stronger result, namely that $\mathbf{Bad}(s,t)$ is hyperplane absolute winning in the sense of Broderick, Fishman, Kleinbock, Reich, and Weiss (2012). As a consequence one can deduce the full dimension of $\mathbf{Bad}(s,t)$ intersected with certain fractals.

Monodromy of Hyperplane Sections of Curves and Decomposition Statistics over Finite Fields

For a projective curve $C\subset\mathbf{P}^n$ defined over $\mathbf{F}_q$ we study the statistics of the $\mathbf{F}_q$-structure of a section of $C$ by a random hyperplane defined over $\mathbf{F}_q$ in the $q\to\infty$ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.

On primitive elements of finite fields avoiding affine hyperplanes

Let $n\ge 2$ be an integer and let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given $\mathbb F_q$-affine hyperplanes $\mathcal A_1, \ldots, \mathcal A_n$ of $\mathbb F_{q^n}$ in general position, we study the existence and distribution of primitive elements of $\mathbb F_{q^n}$, avoiding each $\mathcal A_i$. We obtain both asymptotic and concrete results, relating to past works on digits over finite fields.

$\mathbf{Bad}(\mathbf{r};\mathbf{s})$ 是超平面绝对赢集

$\mathbf{Bad}(\mathbf{r};\mathbf{s})$ is Hyperplane Absolute Winning

本文证明给定维数权向量下坏逼近矩阵集合是超平面绝对赢集，完全解决了 Kleinbock 1998 年的一个开放问题。

有限域中避开仿射超平面的特殊类型本原对的存在性

Existence of Special Types Primitive Pairs in Finite Fields Avoiding Affine Hyperplanes

该研究针对阶为素数幂$q$的$m$次扩域有限域，探讨满足一般位置的仿射超平面规避条件、且元素与其二次多项式像均为本原元的本原对存在性问题，针对高阶域给出了存在性结论。

Bad($\mathbf{w}$) is hyperplane absolute winning

In 1998 Kleinbock conjectured that any set of weighted badly approximable $d\times n$ real matrices is a winning subset in the sense of Schmidt's game. In this paper we prove this conjecture in full for vectors in $\mathbf{R}^d$ in arbitrary dimensions by showing that the corresponding set of weighted badly approximable vectors is hyperplane absolute winning. The proof uses the Cantor potential game played on the support of Ahlfors regular absolutely decaying measures and the quantitative non-divergence estimate for a class of fractal measures due to Kleinbock, Lindenstrauss and Weiss. To esta…