Related papers: A Khintchine type theorem for hyperplanes

We give an affine proof of Feuerbach's theorem, by constructing an explicit affine map which takes the nine-point circle of any given Euclidean triangle to the incircle and fixes the Feuerbach point. The proof is shown to be valid in any…

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and…

Let \Gamma be a geometrically finite tree lattice. We prove a Khintchine-Sullivan type theorem for the Hausdorff measure of the points at infinity of the tree that are well approximated by the parabolic fixed points of G. Using Bruhat-Tits…

Recently, Ghosh \& Haynes \cite{HG} proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarn\'{\i}k-type result also holds…

Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that…

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for every holomorphic map f from the unit disc…

Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended…

We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a…

Given a family of complex affine planes, we show that it is trivial over a Zariski open subset of the base. The proof relies upon a relative version of the contraction theorem.

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric…

In this paper we improve estimates of Jarnik and Apfelbeck for uniform Diophantine exponents of transposed systems of linear forms and generalize to the case of an arbitrary system the estimates of Laurent and Bugeaud for individual…

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an…

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence…

By means of fixed point index theory for multi-valued maps, we provide an analogue of the classical Birkhoff--Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general…

We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex…

The inhomogeneous metric theory for the set of simultaneously $\psi$-approximable points lying on a planar curve is developed. Our results naturally incorporate the homogeneous Khintchine-Jarnik type theorems recently established in [Ann.…

In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic…

We prove a Heinz type inequality for harmonic diffeomorphisms of of the half-plane onto itself. We then apply this result to prove some sharp bound of the Gaussian curvature of a minimal surface, provided that it lies above the whole…

Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for…