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Pankaj Vishe

Publications and source records attributed to Pankaj Vishe.

17 recordsLinked to original sources
arXiv2026

The Fully Inhomogeneous $p$-Adic Littlewood Conjecture

We prove that for almost every $α\in\RR$, the \textit{fully inhomogeneous $p$-adic Littlewood Conjecture} holds; namely, \begin{equation*} \forall δ\in\RR,\forall κ\in\ZZ_p, \;\;\liminf_{\av{q}\rightarrow+\infty}q\inn{qα+δ}\abs{q+κ}_p=0. \end{equation*} Here, $\inn{\cdot}$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic norm. Moreover, we prove this conjecture for every quadratic irrational $α$. This gives an affirmative answer to the $p$-adic version of a question posed by Cassels in \cite{Cas59}.

math.DS↗
arXiv2026

A two-dimensional delta symbol method and its application to pairs of quadratic forms

We present a two-dimensional delta symbol method that facilitates a version of the Kloosterman refinement of the circle method, addressing a question posed by Heath-Brown. As an application, we establish the asymptotic formula for the number of integral points on a non-singular intersection of two integral quadratic forms with at least $10$ variables. Assuming the Generalized Lindelöf Hypothesis, we reduce the number of variables to $9$ by performing a double Kloosterman refinement. A heuristic argument suggests our two-dimensional delta symbol will typically outperform known expressions of this type by an increasing margin as the number of variables grows.

math.NT↗
arXiv2026

Effective equidistribution of unipotent orbits in homogeneous spaces of $\SL(2,\R)\ltimes(\R^2)^{k}$

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $Γ$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially effective asymptotic equidistribution results for expanding translates of $u_{\R}$-orbits and for long pieces of individual $u_{\R}$-orbits in $Γ\backslash G$. An important ingredient of the proof is the delta symbol version of the circle method.

math.DS↗
arXiv2024

Rational curves on complete intersections and the circle method

We study the geometry of the space of rational curves on smooth complete intersections of low degree, which pass through a given set of points on the variety. The argument uses spreading out to a finite field, together with an adaptation to function fields of positive characteristic of work by Rydin Myerson on the circle method. Our work also allows us to handle weak approximation for such varieties.

math.AG↗
arXiv2021

On the Hasse principle for complete intersections

We prove the Hasse principle for a smooth projective variety $X\subset \PP^{n-1}_\Q$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\Q$.

math.NT↗
arXiv2020

A sparse equidistribution result for $(\mathrm{SL}(2,\mathbb{R})/Γ_0)^n$

Let $G=\mathrm{SL}(2,\mathbb{R})^n$, let $Γ=Γ_0^n$, where $Γ_0$ is a co-compact lattice in $\mathrm{SL}(2,\mathbb{R})$, let $F(\mathbf{x})$ be a non-singular quadratic form and let $u(x_1,...,x_n)$ denote the unipotent elements in $G$ which generate the standard $n$ dimensional horospherical subgroup, consisting of $2\times 2$ upper triangular unipotent matrices in each co-ordinate. We prove that in absence of any local obstructions for $F$, given any $x_0\in G/Γ$, the sparse subset $\{u(\mathbf{x})x_0:\in\mathbb{Z}^n, F(\mathbf{x})=0\}$ equidistributes in $G/Γ$ as long as $n\geq 481$, independent of the spectral gap of $Γ_0$.

math.DS↗
arXiv2019

Rational points on complete intersections over $\mathbb{F}_q(t)$

A Kloosterman refinement for function fields $K=\mathbb{F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics $X\subset \mathbb{P}^{n-1}_{K}$ , under the assumption that $q$ is odd and $n\geq 9$.

math.NT↗
arXiv2018

An effective equidistribution result for $SL(2,R)\ltimes(R^2)^{\oplus k}$ and application to inhomogeneous quadratic forms

Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $Γ$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $Γ\backslash G$ which project to pieces of closed horocycles in SL$(2,Z)\backslash$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-α)^2+(m_2-β)^2-(m_3-α)^2-(m_4-β)^2$, for $(α,β)$ of Diophantine type, following the approach by Marklof [24] using theta sums.

math.NT↗
arXiv2016

Simultaneous Diophantine approximation - logarithmic improvements

This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.

math.NT↗
arXiv2015

Uniform bounds for period integrals and sparse equidistribution

Let $M=Γ\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$ along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on $M$.

math.DS↗
arXiv2012

A Fast Algorithm to Compute l(1/2, f x χ_q)

Let $f$ be a fixed (holomorphic or Maass) modular cusp form. Let $\cq$ be a Dirichlet character mod $q$. We describe a fast algorithm that computes the value $L(1/2,f\timesχ_q)$ up to any specified precision. In the case when $q$ is smooth or highly composite integer, the time complexity of the algorithm is given by $O(1+|q|^{5/6+o(1)})$.

math.NT↗
arXiv2011

Rapid computation of L-functions for modular forms

Let $f$ be a fixed (holomorphic or Maass) modular cusp form, with $L$-function $L(f,s)$. We describe an algorithm that computes the value $L(f,1/2+ iT)$ to any specified precision in time $O(1+|T|^{7/8})$.

math.NT↗