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Shreyasi Datta

Publications and source records attributed to Shreyasi Datta.

17 recordsLinked to original sources
arXiv2026

On Fourier asymptotics and effective equidistribution

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis. In particular, for any Borel probability measure $μ$, satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehatμ(m)| = O\left(X^{1/2-θ}\right)$ with $θ>7/64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39/64$, as well as a subclass of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $θ$ can be chosen arbitrarily small): there are measures $μ$ with $\widehatμ(ξ) = O\left(|ξ|^{-1/2+ε}\right)$ for which equidistribution fails.

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arXiv2026

Weighted Diophantine approximation on manifolds

We establish a weighted simultaneous Khintchine-type theorem, both convergence and divergence, for all nondegenerate manifolds, which answers a problem posed in [Math. Ann., 337(4):769-796, 2007]. This extends the main results of [Acta Math., 231:1-30, 2023] and [Ann. of Math. (2), 175(1):187-235, 2012] in the weighted set-up. As a by-product of our method, we also obtain a multiplicative Khintchine-type convergence theorem for all nondegenerate manifolds, which is a simultaneous analogue of the celebrated result of Bernik, Kleinbock, and Margulis for dual approximation.

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arXiv2025

Rational points near manifolds and Khintchine theorem

In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the Khintchine type theorem proved in [Ann. of Math. 175 (2012), 187-235]. Furthermore, we obtain a Jarník-type refinement of our main result, which uses Hausdorff measures. The results are also obtained in the inhomogeneous setting. The proofs are underpinned by a sharper version of a `quantitative nondivergence' result of Bernik--Kleinbock--Margulis and a duality argument which we use to study rational points near manifolds.

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arXiv2025

Winning and nullity of inhomogeneous bad

We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in $\mathbb{R}^d$. This answers a question by Beresnevich--Nesharim--Yang and extends the main result of [Geometric and Functional Analysis, 31 (1), 1-33, 2021] to the inhomogeneous set-up. We also show for any nondegenerate curve and nondegenerate analytic manifold that almost every point is not weighted inhomogeneous badly approximable for any weight. This is achieved by duality and the quantitative nondivergence estimates from homogeneous dynamics motivated by [Acta Math. 231 (2023), 1-30], together with the methods from [arXiv:2307.10109].

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arXiv2024

On weighted singular vectors for multiple weights

We introduce the notion of weighted singular vectors and weighted uniform exponent with respect to a set of weights. We prove invariance of these exponents for affine subspaces and submanifolds inside those affine subspaces. For certain analytic submanifolds, we show that there are totally irrational vectors with high weighted uniform exponent, extending the previously known existence results. Moreover, we show existence and non-existence of non-obvious divergence orbits for certain cones.

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arXiv2024

Winning of inhomogeneous bad for curves

We prove the absolute winning property of weighted simultaneous inhomogeneous badly approximable vectors on non-degenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an inhomogeneous version of the main result in \cite{BNY22} by Beresnevich, Nesharim, and Yang. Also, the generality of the inhomogeneous part that we considered extends the previous result in \cite{ABV}. Moreover, our results even contribute to classical results, namely establishing the inhomogeneous Schmidt's conjecture in arbitrary dimensions.

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arXiv2024

Good functions, measures, and the Kleinbock-Tomanov conjecture

In this paper we prove a conjecture of Kleinbock and Tomanov \cite[Conjecture~FP]{KT} on Diophantine properties of a large class of fractal measures on $\mathbb{Q}_p^n$. More generally, we establish the $p$-adic analogues of the influential results of Kleinbock, Lindenstrauss, and Weiss \cite{KLW} on Diophantine properties of friendly measures. We further prove the $p$-adic analogue of one of the main results in \cite{Kleinbock-exponent} due to Kleinbock concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of \cite{KLW} is a result on $(C, α)$-good functions whose proof crucially uses the mean value theorem. Our main technical innovation is an alternative approach to establishing that certain functions are $(C, α)$-good in the $p$-adic setting. We believe this result will be of independent interest.

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arXiv2023

On absolutely friendly measures on $\mathbb{Q}_S^d$

In this paper, we extend the work of Pollington and Velani in \cite{PV} to an $S$-arithmetic set-up, where $S$ is a finite set of valuations of $\mathbb{Q}$. In particular, for an `absolutely friendly' measure supported on a compact set in $\mathbb{Q}_S^d$, we give a summation condition on approximating function $ψ$ such that $μ$ almost no point in the compact set is $ψ$ approximable. The crucial ingredient is a version of the simplex lemma that we prove dynamically.

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arXiv2023

Bad is null

In this paper we develop a general framework of badly approximable points in a metric space $X$ equipped with a $σ$-finite doubling Borel regular measure $μ$. We establish that under mild assumptions the $μ$-measure of the set of badly approximable points is always zero. The framework can be applied to a variety of settings in Diophantine approximation and dynamical systems, which we also consider, including weighted and $S$-arithmetic Diophantine approximations, Diophantine approximation on manifolds and intrinsic approximations on fractals.

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arXiv2023

Rectangular shrinking targets for $\mathbb{Z}^m$ actions on tori: well and badly approximable systems

In this paper we investigate the shrinking target property for irrational rotations. This was first studied by Kurzweil (1951) and has received considerable interest of late. Using a new approach, we generalize results of Kim (2007) and Shapira (2013) by proving a weighted effective analogue of the shrinking target property. Furthermore, our results are established in the much wider $S$-arithmetic setting.

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arXiv2022

Quantitative Khintchine in Simultaneous Approximation

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 approximation. In this paper, we prove an effective version of their result.

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arXiv2022

Singular Vectors in Real Affine Subspaces

We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside those affine subspaces. We define a notion of $n$-singularity for matrices, which is closely related to the uniform exponent of irrationality. For certain affine subspaces, we show that the set of singular vectors has measure zero if and only if the parametrizing matrix is not $n$-singular. In particular, we show for affine hyperplanes the set of singular vectors has measure zero if and only if the parametrizing matrix is not rational.

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arXiv2022

Singular Vectors and $ψ$-Dirichlet Numbers over Function Field

We show that the only $ψ$-Dirichlet numbers in a function field over a finite field are rational functions, unlike $ψ$-Dirichlet numbers in $\mathbb{R}$. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field.

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arXiv2022

Singular Vectors on Manifolds over totally real Number Fields

We extend the notion of singular vectors in the context of Diophantine approximation of real numbers with elements of a totally real number field $K$. For $m\geq1$, we establish a version of Dani's correspondence in number fields and prove that under a class of `friendly measures' in $K_S^m$, the set of singular vectors has measure zero. Here $S$ is the set of Archimedean valuations of $K$ and $K_S$ is the product of the completions of $σ(K)$, $σ\in S$. On the other hand, we show the existence of uncountably many non-trivial singular vectors on suitable submanifolds of $K^m_S$ under the action of a certain one parameter subgroup of $\mathrm{SL}_{m+1}(K_S)$.

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arXiv2020

$S$-arithmetic Inhomogeneous Diophantine approximation on manifolds

We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For $S$ consisting of more than one prime, finite or infinite, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprindžuk conjectures using nondivergence estimates for flows on homogeneous spaces in conjunction with the transference principle of Beresnevich and Velani.

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arXiv2019

Diophantine inheritance for p-adic measures

In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a stronger form), a conjecture of Kleinbock and Tomanov \cite{KT}, as well as a question of Kleinbock \cite{Kleinbock-exponent}. Our main innovation is the introduction of a new $p$-adic Diophantine exponent which is better suited to homogeneous dynamics, and which we show to be closely related to the exponent considered by Kleinbock and Tomanov.

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