Extremal subspaces and their submanifolds

极值仿射子空间与Khintchine-Jarník型定理

Extremal affine subspaces and Khintchine-Jarník type theorems

本文证明Kleinbock猜想，完整分类极值仿射子空间，并给出Khintchine型及Jarník型定理的分类，基于有理点计数新估计，运用多维大筛不等式。

Diophantine Exponents of Affine Subspaces: The Simultaneous Approximation Case

We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.

Singular Vectors in Real Affine Subspaces

We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside those affine subspaces. We define a notion of $n$-singularity for matrices, which is closely related to the uniform exponent of irrationality. For certain affine subspaces, we show that the set of singular vectors has measure zero if and only if the parametrizing matrix is not $n$-singular. In particular, we show for affine hyperplanes the set of singular vectors has measure zero if and only if the parametrizing matrix is not rational.

关于多权重的加权奇异向量

On weighted singular vectors for multiple weights

该研究引入多权重下的加权奇异向量与加权一致指数概念，证明其仿射子空间相关不变性，拓展高指数完全无理向量的存在性结果，并明确特定锥下非平凡发散轨道的存在与不存在情况。

Freiman's theorem in finite fields via extremal set theory

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asyptotically sharp version of Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A| then A is contained in a subspace of size 2^{2K + O(\sqrt{K}\log K)}|A|; except for the O(\sqrt{K} \log K) error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K^{46}) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman-Ruzsa conjecture in this case. A common theme i…

Baker-Sprindzhuk conjectures for complex analytic manifolds

We show a large class of analytic submanifolds of C^n to be strongly extremal. This generalizes V. Sprindzhuk's solution of the complex case of Mahler's Problem, and settles complex analogues of conjectures made in the 1970s by Baker and Sprindzhuk. The proof is based on a variation of quantitative nondivergence estimates for quasi-polynomial flows on the space of lattices.