arXiv subjects

Gaurav Aggarwal

Publications and source records attributed to Gaurav Aggarwal.

At least 19 recordsLinked to original sources

arXiv2026

Cone upgrade for effective equidistribution and weighted Khintchine--Schmidt theorem on fractals

We prove a fractal analogue of the weighted theorems of Khintchine and Schmidt for products of self-similar measures on the real line whose defining iterated function systems share a common contraction ratio. The proof translates the counting problem into homogeneous dynamics and builds on recent work of B\'enard, He and Zhang, who treated the unweighted case. Our main new ingredient is a \emph{cone upgrade}: an effective double equidistribution theorem for translates of product self-similar measures, uniform over diagonal elements whose weights range over a fixed cone in the positive Weyl chamber, deduced from effective equidistribution for a single weight. This yields a partial effective form of a theorem of Khalil, Luethi and Weiss.

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arXiv2026

Generalised Gauss--Kuzmin Distribution for Klein sails in $\mathbb R^3$

The classical Gauss--Kuzmin distribution describes the asymptotic distribution of continued fraction digits. Geometrically, this may be interpreted as the equidistribution of faces of Klein sails in $\mathbb R^2$. In this paper, we establish a three-dimensional analogue of this phenomenon. We prove the equidistribution of local face structures in generic Klein sails in $\mathbb R^3$, thereby obtaining a higher-dimensional generalization of the Gauss--Kuzmin distribution. In addition, we resolve several open questions posed by Karpenkov~\cite{Ka17}. Our approach is based on homogeneous dynamics and is motivated from the work of Kontsevich and Suhov~\cite{KS99}. More precisely, we construct a cross-section for the diagonal flow on $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, such that visits to the cross-section encode the geometry of three-dimensional sails. The principal technical contribution is the proof that the associated cross-sectional measure is finite, which enables us to derive the limiting face statistics.

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arXiv2026

Fine-scale statistics for $\mathbb{Q}^n$

We study the distribution of rational points in $\mathbb{R}^n$, with denominators restricted to the interval $[Q-\Delta, Q]$, and $Q,\Delta\to\infty$ such that $\Delta/Q\to 0$. Previous results in the literature, due to Hall and others, were limited to Farey sequences, where the window size $\Delta$ is of the same order as $Q$. We prove the convergence of fine-scale statistics in a range of scaling limits and express the limit laws in terms of natural probability measures on the space of affine lattices. The key technical ingredient of our approach is an equidistribution theorem for slowly expanding horospheres, with some new exotic limit measures. Our techniques furthermore allow us to answer a recent question by Anderson, Boca, Cobeli and Zaharescu concerning the directional statistics of lattice points.

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arXiv2026

Effective multi-equidistribution for translates of unipotent flows and Central limit theorems in inhomogeneous Diophantine approximation

In this paper, we prove a central limit theorem for inhomogeneous Diophantine approximation with a fixed shift, provided the shift is non-Liouville. This generalizes earlier work of Dolgopyat, Fayad, and Vinogradov~\cite{DFV}. This is achieved by translating the problem to one involving flows on homogeneous spaces. In this latter setting, we establish an effective multi-equidistribution result for diagonal translates of unipotent flows. This result is obtained by combining a recent result of Kim~\cite{Kim2024} with the height function construction of Shi~\cite{Shi20}. The central limit theorem is then deduced using the method of Bj\"orklund and Gorodnik~\cite{BG}.

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arXiv2025

The Doeblin-Lenstra conjecture: effective results and central limit theorems

We establish effective convergence rates in the Doeblin-Lenstra law, describing the limiting distribution of approximation coefficients arising from continued fraction convergents of a typical real number. More generally, we prove quantitative versions of the Doeblin-Lenstra law for best approximates in higher dimensions, as well as for points sampled from fractal measures on the real line, including the middle-third Cantor measure. Our method reduces the problem to proving effective convergence of Birkhoff averages for diagonal flows on spaces of unimodular lattices. A key step is to show that, despite the discontinuity of the observable of interest, it satisfies the regularity conditions on average required for effective ergodic theorems. For the fractal setting, we establish effective multi-equidistribution properties of self-similar measures under diagonal flow, extending recent work on single equidistribution by B\'enard, He and Zhang. As a consequence, we also obtain central limit theorems for these Diophantine statistics in both classical and fractal settings.

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arXiv2025

Farm-Level, In-Season Crop Identification for India

Accurate, timely, and farm-level crop type information is paramount for national food security, agricultural policy formulation, and economic planning, particularly in agriculturally significant nations like India. While remote sensing and machine learning have become vital tools for crop monitoring, existing approaches often grapple with challenges such as limited geographical scalability, restricted crop type coverage, the complexities of mixed-pixel and heterogeneous landscapes, and crucially, the robust in-season identification essential for proactive decision-making. We present a framework designed to address the critical data gaps for targeted data driven decision making which generates farm-level, in-season, multi-crop identification at national scale (India) using deep learning. Our methodology leverages the strengths of Sentinel-1 and Sentinel-2 satellite imagery, integrated with national-scale farm boundary data. The model successfully identifies 12 major crops (which collectively account for nearly 90% of India's total cultivated area showing an agreement with national crop census 2023-24 of 94% in winter, and 75% in monsoon season). Our approach incorporates an automated season detection algorithm, which estimates crop sowing and harvest periods. This allows for reliable crop identification as early as two months into the growing season and facilitates rigorous in-season performance evaluation. Furthermore, we have engineered a highly scalable inference pipeline, culminating in what is, to our knowledge, the first pan-India, in-season, farm-level crop type data product. The system's effectiveness and scalability are demonstrated through robust validation against national agricultural statistics, showcasing its potential to deliver actionable, data-driven insights for transformative agricultural monitoring and management across India.

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arXiv2025

L\'{e}vy-Khintchine Theorems: effective results and central limit theorems

The L\'evy-Khintchine theorem is a classical result in Diophantine approximation that describes the asymptotic growth of the denominators of convergents in the continued fraction expansion of a typical real number. An effective version of this theorem was proved by Phillip and Stackelberg (\textit{Math. Annalen}, 1969) and Central Limit Theorems were proved by several authors \cites{Ibragimov, Misevicius, Morita, Vallee}. In this work, we develop a new approach towards quantifying the L\'evy-Khintchine theorem. Our methods apply to the setting of higher-dimensional simultaneous Diophantine approximation, thereby providing an effective version of a theorem of Cheung and Chevallier (\textit{Annales scientifiques de l'ENS}, 2024). Further, we prove a Central Limit Theorem for best approximations in all dimensions. Unlike previous approaches to the one-dimensional problem, our approach relies on techniques from homogeneous dynamics.

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arXiv2025

Dimension bounds for singular affine forms

In this paper, we establish upper bounds on the dimension of sets of singular-on-average and $\omega$-singular affine forms in singly metric settings, where either the matrix or the shift is fixed. These results partially address open questions posed by Das, Fishman, Simmons, and Urba\'nski, as well as Kleinbock and Wadleigh. Furthermore, we extend our results to the generalized weighted setup and derive bounds for the intersection of these sets with a wide class of fractals.

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arXiv2024

On the packing dimension of weighted singular matrices on fractals

We provide the first known upper bounds for the packing dimension of weighted singular and weighted $\omega$-singular matrices. We also prove upper bounds for these sets when intersected with fractal subsets. The latter results, even in the unweighted setting, are already new for matrices. Further, even for row vectors, our results enlarge the class of fractals for which bounds are currently known. We use methods from homogeneous dynamics, in particular we provide upper bounds for the packing dimension of points on the space of unimodular lattices, whose orbits under diagonal flows $p$-escape on average.

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arXiv2024

Agricultural Landscape Understanding At Country-Scale

Comprehensive agricultural landscape understanding is critical for addressing global challenges in food security, climate change, and resource management. This requires mapping not just crop fields, but also vital features like trees and water bodies which form an intricate mosaic in complex \textit{smallholder} systems dominating the Global South. Previous efforts to develop such land use maps have been limited by a narrow focus on methods for field delineation only, and also do not develop robust post-processing steps essential for real-world deployment. Further, to our knowledge, no prior system for smallholder farms has been deployed and evaluated at a national scale. This work addresses these limitations by presenting the first national-scale agricultural mapping system that moves beyond simple field delineation to enable segmentation of agricultural instances like fields, trees and water bodies. Our system is refined for real-world application using novel post-processing heuristics to ensure map consistency and accuracy, and is validated through a rigorous, multi-faceted evaluation process. Fine-grained land use maps generated by our system are publicly accessible via an API at \textit{\href{http://agri.withgoogle.com}{http://agri.withgoogle.com}}, enabling a wide range of applications from precision agriculture and policy-making to advancing global sustainability development goals.

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arXiv2024

On the Hausdorff dimension of Weighted Singular Vectors

We prove a sharp upper bound on the Hausdorff dimension of weighted singular vectors in $\mathbb{R}^m$ using dynamics on homogeneous spaces, specifically the method of integral inequalities. Together with the lower bound proved recently by Kim and Park \cite{KimPark2024}, this determines the Hausdorff dimension of weighted singular vectors, thereby generalizing to arbitrary dimension, the work of Liao, Shi, Solan, and Tamam \cite{LSST}, who determined the Hausdorff dimension of weighted singular vectors in two dimensions. We also provide the first known bounds for the Hausdorff dimension of weighted singular vectors restricted to fractal subsets.

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arXiv2024

Efficient Data Subset Selection to Generalize Training Across Models: Transductive and Inductive Networks

Existing subset selection methods for efficient learning predominantly employ discrete combinatorial and model-specific approaches which lack generalizability. For an unseen architecture, one cannot use the subset chosen for a different model. To tackle this problem, we propose $\texttt{SubSelNet}$, a trainable subset selection framework, that generalizes across architectures. Here, we first introduce an attention-based neural gadget that leverages the graph structure of architectures and acts as a surrogate to trained deep neural networks for quick model prediction. Then, we use these predictions to build subset samplers. This naturally provides us two variants of $\texttt{SubSelNet}$. The first variant is transductive (called as Transductive-$\texttt{SubSelNet}$) which computes the subset separately for each model by solving a small optimization problem. Such an optimization is still super fast, thanks to the replacement of explicit model training by the model approximator. The second variant is inductive (called as Inductive-$\texttt{SubSelNet}$) which computes the subset using a trained subset selector, without any optimization. Our experiments show that our model outperforms several methods across several real datasets

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arXiv2024

Generalized L\'{e}vy-Khintchine Theorems and a Conjecture of Y. Cheung

The celebrated L\'evy--Khintchine theorem is a fundamental limiting law that describes the growth rate of the denominators of the convergents in the continued fraction expansion of a Lebesgue-typical real number. In a recent breakthrough, Cheung and Chevallier \textit{(Annales scientifiques de l'ENS, 2024)} extended this theorem to higher dimensions. In this paper, we resolve a conjecture of Y. Cheung and answer a question of Cheung and Chevallier concerning L\'evy--Khintchine type theorems for arbitrary norms. We also establish a higher-dimensional analogue of the Doeblin--Lenstra law. While our results are new in higher dimensions, they also yield significant improvements in the classical one-dimensional setting. Specifically, we revisit the L\'evy--Khintchine theorem and the Doeblin--Lenstra law through the lens of Mahler's influential proposal to study Diophantine approximation on fractals. In particular, we prove these results for almost every point on the middle-third Cantor set. More broadly, our framework applies to a wide class of measures, including those supported on curves and on self-similar fractals generated by iterated function systems (IFS), and it also allows constraints on the selection of best approximates.

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arXiv2024

Non-Expanding Random walks on Homogeneous spaces and Diophantine approximation

We study non-expanding random walks on the space of affine lattices and establish a new classification theorem for stationary measures. Further, we prove a theorem that relates the genericity with respect to these random walks to Birkhoff genericity. Finally, we apply these theorems to obtain several results in inhomogeneous Diophantine approximation, especially on fractals.

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arXiv2024

Counting and Joint equidistribution of approximates

In this paper, we consider the problem of counting Diophantine inequalities with multiple natural constraints. We prove a very general result in this setting using dynamical techniques. More precisely, we consider the joint asymptotic distribution of $\varepsilon$-Diophantine approximates of matrices in several aspects. Our main results describe the resulting limiting measures for almost every matrix. Multiplicative Diophantine approximation is treated for the first time, and a number of Diophantine corollaries are derived. While we treat the general case of approximation of matrices, our results are already new for the case of simultaneous Diophantine approximation of vectors. Our approach is dynamical and is based on the construction of an appropriate Poincar\'{e} section for certain diagonal group actions on the space of unimodular lattices along with multiple mixing. The main new idea in our paper is a method that allows us to treat actions of higher rank groups.

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arXiv2023

Analyzing the Efficacy of an LLM-Only Approach for Image-based Document Question Answering

Recent document question answering models consist of two key components: the vision encoder, which captures layout and visual elements in images, and a Large Language Model (LLM) that helps contextualize questions to the image and supplements them with external world knowledge to generate accurate answers. However, the relative contributions of the vision encoder and the language model in these tasks remain unclear. This is especially interesting given the effectiveness of instruction-tuned LLMs, which exhibit remarkable adaptability to new tasks. To this end, we explore the following aspects in this work: (1) The efficacy of an LLM-only approach on document question answering tasks (2) strategies for serializing textual information within document images and feeding it directly to an instruction-tuned LLM, thus bypassing the need for an explicit vision encoder (3) thorough quantitative analysis on the feasibility of such an approach. Our comprehensive analysis encompasses six diverse benchmark datasets, utilizing LLMs of varying scales. Our findings reveal that a strategy exclusively reliant on the LLM yields results that are on par with or closely approach state-of-the-art performance across a range of datasets. We posit that this evaluation framework will serve as a guiding resource for selecting appropriate datasets for future research endeavors that emphasize the fundamental importance of layout and image content information.

cs.CV↗

arXiv2023

Is it an i or an l: Test-time Adaptation of Text Line Recognition Models

Recognizing text lines from images is a challenging problem, especially for handwritten documents due to large variations in writing styles. While text line recognition models are generally trained on large corpora of real and synthetic data, such models can still make frequent mistakes if the handwriting is inscrutable or the image acquisition process adds corruptions, such as noise, blur, compression, etc. Writing style is generally quite consistent for an individual, which can be leveraged to correct mistakes made by such models. Motivated by this, we introduce the problem of adapting text line recognition models during test time. We focus on a challenging and realistic setting where, given only a single test image consisting of multiple text lines, the task is to adapt the model such that it performs better on the image, without any labels. We propose an iterative self-training approach that uses feedback from the language model to update the optical model, with confident self-labels in each iteration. The confidence measure is based on an augmentation mechanism that evaluates the divergence of the prediction of the model in a local region. We perform rigorous evaluation of our method on several benchmark datasets as well as their corrupted versions. Experimental results on multiple datasets spanning multiple scripts show that the proposed adaptation method offers an absolute improvement of up to 8% in character error rate with just a few iterations of self-training at test time.

cs.CV↗

arXiv2023

Weakly supervised information extraction from inscrutable handwritten document images

State-of-the-art information extraction methods are limited by OCR errors. They work well for printed text in form-like documents, but unstructured, handwritten documents still remain a challenge. Adapting existing models to domain-specific training data is quite expensive, because of two factors, 1) limited availability of the domain-specific documents (such as handwritten prescriptions, lab notes, etc.), and 2) annotations become even more challenging as one needs domain-specific knowledge to decode inscrutable handwritten document images. In this work, we focus on the complex problem of extracting medicine names from handwritten prescriptions using only weakly labeled data. The data consists of images along with the list of medicine names in it, but not their location in the image. We solve the problem by first identifying the regions of interest, i.e., medicine lines from just weak labels and then injecting a domain-specific medicine language model learned using only synthetically generated data. Compared to off-the-shelf state-of-the-art methods, our approach performs >2.5x better in medicine names extraction from prescriptions.

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