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Arijit Ganguly

Publications and source records attributed to Arijit Ganguly.

12 recordsLinked to original sources
arXiv2025

Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case

We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the method of counting rational points near manifolds developed by Beresnevich and Yang. To address the scenario where the given approximating function is not monotonic, we extend our function field by adjoining an appropriate root. Additionally, in the course of the proof, we establish several new results in the geometry of numbers over function fields, which are of independent interest.

math.NT↗
arXiv2023

Diophantine transference principle over function fields

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 inhomogeneous Diophantine approximation over function fields. This also yields the inhomogeneous Baker-Sprindzuk conjecture over function fields as a consequence. Furthermore, we prove the upper bounds for the general non-extremal scenario.

math.NT↗
arXiv2022

Inhomogeneous Khitchine-Groshev type theorems on manifolds over function fields

The goal of this paper is to establish a complete Khintchine-Groshev type theorem in both homogeneous and inhomogeneous setting, on analytic nondegenerate manifolds over a local field of positive characteristic. The dual form of Diophantine approximation has been considered here. Our treatise provides the function field analogues of the various earlier results of this type, studied in the euclidean and S-adic framework, by Bernik, Kleinbock and Margulis, Beresnevich, Bernik, Kleinbock and Margulis, Badziahin, Beresnevich and Velani, Mohammadi and Golsefidy, and Datta and Ghosh.

math.NT↗
arXiv2022

Random walks on tori and normal numbers in self similar sets

We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \subset \mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x \mapsto D^{-r_i} x + t_i \ (i=1, \ldots, n)$, where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, and $r_{i} \in\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx$ (multiplication mod $\mathbb{Z}^{d}$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base 3.

math.DS↗
arXiv2022

Dirichlet improvability for $S$-numbers

We study the problem of improving Dirichlet's theorem of metric Diophantine approximation in the $S$-adic setting. Our approach is based on translation of the problem related to Dirichlet improvability into a dynamical one, and the main technique of our proof is the $S$-adic version of quantitative nondivergence estimate due to D. Y. Kleinbock and G. Tomanov. The main result of this paper can be regarded as the number field version of earlier works of D. Y. Kleinbock and B. Weiss, and of the second named author and Anish Ghosh. Also this in turn generalises a result of Shreyasi Datta and M. M. Radhika on singularity of vectors to any number field $K$ and $S$ containing all archimedian places.

math.NT↗
arXiv2021

Khintchine's theorem for affine hyperplanes in positive charateristic

In this paper we establish the convergence case of Khintchine's theorem for affine hyperplanes in function field of positive characteristic. Along with that, we also prove a quantitative version of the same. The main technique used in the proof is a dynamical result called `Quantitative nondivergence' due to D. Y. Kleinbock and G. A. Margulis [17].

math.NT↗
arXiv2017

Inhomogeneous dual Diophantine approximation on affine subspaces

We prove the convergence and divergence cases of an inhomogeneous Khintchine-Groshev type theorem for dual approximation restricted to affine subspaces in $\mathbb{R} ^n$. The divergence results are proved in the more general context of Hausdorff measures.

math.NT↗
arXiv2016

Quantitative Diophantine approximation on affine subspaces

Recently, Adiceam, Beresnevich, Levesley, Velani and Zorin proved a quantitative version of the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds, motivated by applications to interference alignment. In the present paper, we obtain analogues of their results for affine subspaces.

math.NT↗
arXiv2015

Dirichlet's Theorem in function fields

We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet's theorem in Diophantine approximation and prove very general results in this context.

math.NT↗
arXiv2006

IP over P2P: Enabling Self-configuring Virtual IP Networks for Grid Computing

Peer-to-peer (P2P) networks have mostly focused on task oriented networking, where networks are constructed for single applications, i.e. file-sharing, DNS caching, etc. In this work, we introduce IPOP, a system for creating virtual IP networks on top of a P2P overlay. IPOP enables seamless access to Grid resources spanning multiple domains by aggregating them into a virtual IP network that is completely isolated from the physical network. The virtual IP network provided by IPOP supports deployment of existing IP-based protocols over a robust, self-configuring P2P overlay. We present implementation details as well as experimental measurement results taken from LAN, WAN, and Planet-Lab tests.

cs.DC↗