Related papers: Finiteness Problems in Diophantine Geometry
This paper is a survey of old and recent results related to Khintichine's singular matrices and their applications in the theory of Diophantine approximations. The paper is written in Russian. English version should appear in "Russian…
Presentaremos una nueva demostraci\'on del teorema de Shafarevich sobre finitud de curvas el\'ipticas con buena reducci\'on fuera de un conjunto finito de primos dado. Esto da un nuevo punto de entrada a teoremas fundamentales de finitud…
We comment on the paper \"Uber Extremalprobleme der konformen Geometrie (On extremal problems in conformal geometry) by Teichm\"uller, published in 1941. This paper contains ideas on a wide generalization of his previous work on the…
I present here the proofs of results, which are obtained in my papers "On the linear forms with algebraic coefficoients of logarithms of algebraic numbers", VINITI, 1996, 1617-B96, pp. 1 - 23 (in Russian), and "On the systems of linear…
In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms…
This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.
We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.
We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine $m$-tuples in finite fields, which is due to A. Dujella and M.Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).
In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various…
In 1979, James B.~Saxe published an extended summary on the complexity of the Distance Geometry Problem in the proceedings of the 17th Allerton Conference. Many of the proofs in his paper are sketches, and even the whole proofs do not have…
This text is the write-up of a talk at the Bellairs Workshop in Number Theory on Tropical and Non-Archimedean Geometry that took place at the Bellairs Research Institute, Barbados, in May 2011. The goal of this text is to present recent…
Between his arrival in Frankfurt in $1922$ and and his proof of his famous finiteness theorem for integral points in $1929$, Siegel had no publications. He did, however, write a letter to Mordell in $1926$ in which he explained a proof of…
In this paper, we introduce an algebro-geometric formulation for Faltings' theorem on diophantine approximation on abelian varieties using an improvement of Faltings-Wustholz observation over number fields. In fact, we prove that, for any…
This is a (mostly expository) paper on Reidemeister classes, twisted Burnside-Frobenius theory, congruences, R-infinity property and all that. It was written in 2005 and published in 2008. We post it as it was, only the bibliography data is…
We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine…
The paper is devoted to the study of homeomoephisms with finite distortion on the plane with use of the modulus techniques.
This article is a translation of Michael Sadowsky's original paper "Ein elementarer Beweis f\"ur die Existenz eines abwickelbaren M\"obiusschen Bandes und die Zur\"uckf\"uhrung des geometrischen Problems auf ein Variationsproblem." which…
This is a significant revision of the early version of this paper which was posted last December. The speculative section has been removed in light of some recent results of Morita and Kawazumi. Numerous typos have been fixed. The companion…
In a recent paper, J.-B. Bost establishes a criterion for certain ``formal subvarieties'' of algebraic varieties to be algebraic. His theorem unifies and generalizes results of Chudnovsky's and Y. Andr\'e, motivated by an arithmetic…
The objective of this paper is to (partially) address the issue of finding an analogue to Khintchine's theorem for IFS Fractals. We study the convergence case for Diophantine approximations, and show an improved result for higher…