• Corpus ID: 248572539

On the generalisation of Roth's theorem

  title={On the generalisation of Roth's theorem},
  author={Paolo Dolce and Francesco Zucconi},
We present two possible generalisations of Roth’s approximation theorem on proper adelic curves, assuming some technical conditions on the behavior of the logarithmic absolute values. We illustrate how tightening such assumptions makes our inequalities stronger. As special cases we recover Corvaja’s results [Cor97] for fields admitting a product formula, and Vojta’s ones [Voj21] for arithmetic function fields. the following: 


Some quantitative results related to Roth's Theorem
  • A. J. Poorten
  • Mathematics
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
  • 1988
Abstract We employ the Dyson's Lemma of Esnault and Viehweg to obtain a new and sharp formulation of Roth's Theorem on the approximation of algebraic numbers by algebraic numbers and apply our
Autour du théorème de Roth
On Roth's theorem. The celebrated theorem of Roth, together with its generalizations given by Mahler and Ridout, gives a lower bound for the degree of approximation of one or more algebraic numbers
Rational approximations to algebraic numbers
It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequality has only a finite number of solutions in relatively prime integers p, q ( q > 0) The
Roth’s Theorem over arithmetic function fields
Roth's theorem is extended to finitely generated field extensions of $\Bbb Q$, using Moriwaki's framework for heights.
Arithmetic height functions over finitely generated fields
Abstract.In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s
Analytic Number Theory: The Number of Algebraic Numbers of Given Degree Approximating a Given Algebraic Number
has only finitely many solutions. Roth’s proof is by contradiction. Assuming that (1.1) has infinitely many solutions, Roth constructed an auxiliary polynomial in a large number of variables, k say,
Diophantine Approximations and Diophantine Equations
Siegel's lemma and heights.- Diophantine approximation.- The thue equation.- S-unit equations and hyperelliptic equations.- Diophantine equations in more than two variables.
Heights in Diophantine Geometry
I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil
Arakelov Geometry over Adelic Curves
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
Algebraic Number Theory
I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions