Diophantine Approximation and Lattices with Complex Multiplication.
Diophantine approximation and special Liouville numbers
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers . The approach relies on results on the connection between the set of all -adic expansions () of and their associated approximation constants. As an application, explicit construction of real numbers with prescribed approximation properties are deduced and illustrated by Matlab plots.
Diophantine approximation by square-free numbers
Diophantine approximation, Hausdorff dimension and Schroder's functional equation
Diophantine approximation in Banach spaces
In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.
Diophantine Approximation in Characteristic p.
Diophantine approximation in power series fields
Diophantine approximation in short intervals
Diophantine approximation involving primes.
Diophantine Approximation on Abelian Varieties with Complex Mulitplication.
Diophantine approximation on affine hyperplanes
Diophantine approximation on algebraic varieties
We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.
Diophantine approximation on Veech surfaces
We show that Y. Cheung’s general -continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental...
Diophantine approximation with partial sums of power series
We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.
Diophantine approximation with primes and powers of two.
Diophantine Approximations of Infinite Series and Products
This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos' results on~irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.
Diophantine approximations on projective spaces.
Diophantine approximations related to rational values of G-functions
Diophantine approximations with Fibonacci numbers
Let be the -th Fibonacci number. Put . We prove that the following inequalities hold for any real :1) ,2) ,3) .These results are the best possible.
Diophantine equations after Fermat’s last theorem
These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?