Related papers: A Khintchine type theorem for hyperplanes

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^\alpha\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator…

We study the question of finding smooth hyperplane sections to a pencil of hypersurfaces over finite fields.

We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique K\"ahler-Einstein metric on such a manifold implies uniform Ding stability. The main new…

In this paper, we establish a Schmidt's subspace theorem for moving hyeprplane targets in projective spaces over function fields.

An asymptotic formula which holds almost everywhere is obtained for the number of solutions to the Diophantine inequalities |qA-p|<\psi(|q|), where A is an n by m matrix (m>1) over the field of formal Laurent series with coefficients from a…

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the…

We show that Hilbert schemes for quantum planes are projective.

The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb R^{nm}$.…

These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems…

We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly…

We provide several results on the diophantine properties of continued fractions on the Heisenberg group, many of which are analogous to classical results for real continued fractions. In particular, we show an analog of Khinchin's theorem…

In this paper, we study a point-hyper plane incidence theorem in matrix rings, which generalizes all previous works in literature of this direction.

This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a…

In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem for general (higher) dimensions. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more…

In this article, we give a counterexample to the Lefschetz hyperplane theorem for non-singular quasi-projective varieties. A classical result of Hamm-L\^{e} shows that Lefschetz hyperplane theorem can hold for hyperplanes in general…

In this paper we obtain new inclusion and coincidence theorems for absolutely or multiple summing multilinear mappings. In particular, we derive optimal coincidence theorems of Bohnenblust-Hille type for multilinear forms on K-convex Banach…

An effective algorithm of determining Gromov--Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space is proposed. The Appendix is joint with E. Schulte-Geers.

We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish…

We study the nonresonance phenomenon for complex rank-one local systems on complements of hyperplane arrangements. We refine the method of Cohen, Dimca, and Orlik and obtain a combinatorial sufficient condition for nonresonance. As an…

We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be…