Related papers: On Furstenberg's Diophantine result

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\…

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal…

In this paper, we obtain new observability inequalities for the vibrating string. This work was motivated by a recent paper by A. Szij\'art\'o and J. Heged\H{u}s in which the authors ask the question of determining the initial data by only…

We estalish the conjectures of Sprindzhuk over a local field of positive characteristic using the dynamical method of Kleinbock-Margulis.

In Diophantine approximation, Vaaler's theorem was an important partial result towards the Duffin--Schaeffer conjecture, which was open for almost eighty years before it was recently proven by Koukoulopoulos and Maynard. A version of this…

We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani to the function field set-up, we extend many results from homogeneous Diophantine approximation to the realm of…

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which…

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $\alpha_1x_1+\alpha_2x_2 +y$ with {\it positive} integers $x_1,x_2$.

Aizenbud, Gourevitch, Rallis and Schiffmann proves results of multicity one in different cases, in particular for orthogonal groups. Using the same method, we prove the same result for special orthogonal groups.

Given a monotonically decreasing $\psi: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $\alpha \in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that…

In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\geq 2$ and $y$ are positive integers such that $\gcd(n,d)=1.$

We prove an effective version of the Pila-Wilkie Theorem for sets definable using Pfaffian functions, providing effective estimates for the number of algebraic points of bounded height and degree lying on such sets. We also prove effective…

We find new inequalities between uniform and individual Diophantine exponents for three-dimensional Diophantine approximations. Also we give a result for two linear forms in two variables. The results improves V.Jarnik's theorem (1954).

In a ground-breaking work \cite{BY}, Beresnevich and Yang recently proved Khintchine's theorem in simultaneous Diophantine approximation for nondegenerate manifolds resolving a long-standing problem in the theory of Diophantine…

By applying Schmidt's lattice method, we prove results on simultaneous Diophantine approximation modulo 1 for systems of polynomials in a single prime variable provided that certain local conditions are met.

Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…

We give a simple proof of Bourgain's theorem on the singularity of Ornstein's maps.

We prove an effective estimate for the counting function of Diophantine approximants on the sphere S$^n$. We use homogeneous dynamics on the space of orthogonal lattices, in particular effective equidistribution results and non-divergence…

Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $\Psi$-well approximable sets. These foundational results have since been generalised to the…

In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the…