Related papers: Geometric considerations around the Littlewood con…
We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued…
The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
We establish a near dichotomy between randomness and structure for the point counts of arbitrary projective cubic threefolds over finite fields. Certain "special" subvarieties, not unlike those in the Manin conjectures, dominate. We also…
Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abelian variety is equivalent to a well-known Diophantine…
The Mordell conjecture: origins, approaches, generalizations -- The Mordell conjecture predicts that a diophantine equation defining a smooth projective curve of genus at least two has only finity many solutions in a given number field. The…
In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo absolute value sequence $\mathcal{D}$, we obtain the sharp criterion such that for almost every $\alpha$ the inequality \begin{equation*}…
In this article we present a generalization of a Leibniz's geometrical theorem and an application of it.
We study the dependence on various parameters of the exceptional set in Vojta's conjecture. In particular, by making use of certain elliptic surfaces, we answer in the negative the often-raised question of whether Vojta's conjecture holds…
In this note, we establish the validity of a conjecture recently proposed in Mathematics Magazine and connect it to the existing interesting results
We present some results on projective toric varieties which are relevant in Diophantine geometry. We interpret and study several invariants attached to these varieties in geometrical and combinatorial terms. We also give a B\'ezout theorem…
Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$…
We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.
In this short paper we propose four conjectures in synthetic geometry that generalize Erdos-Mordell Theorem, and three conjectures in number theory that generalize Fermat Numbers.
There is not much that can be said for all $x$ and for all $n$ about the sum \[ \sum_{k=1}^n \frac{1}{|\sin k\pi x|}. \] However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools…
The present paper aims at representing an improvement of the result in [2], where a strong unique continuation property and a description of the local behaviour around the edge of a crack for solutions to an elliptic problem are…
Using contiguous relations we construct an infinite number of continued fraction expansions for ratios of generalized hypergeometric series 3F2(1). We establish exact error term estimates for their approximants and prove their rapid…
We prove a result on the structure of a Diophantine spectrum associated with Minkowski diagonal continued fraction.
We show how to formulate physical theory taking as a starting point the set of states (geometric approach). We discuss the relation of this formulation to the conventional approach to classical and quantum mechanics and the theory of…
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.