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abstract: 'We investigate $S$-arithmetic inhomogeneous Khintchine type theorems in the dual setting for nondegenerate manifolds. We prove the convergence case of the theorem, including, in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprindžuk conjectures. The divergence case is proved for ${\mathbb{Q}}_p$ but in the more general context of Hausdorff measures. This answers a question posed by Badziahin, Beresnevich and Velani [@BaBeVe].'
address: 'School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005, India'
author:
- Shreyasi Datta
- Anish Ghosh
title: '$S$-arithmetic Inhomogeneous Diophantine approximation on manifolds'
---

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Introduction
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In this paper we are concerned with metric Diophantine approximation on nondegenerate manifolds in the $p$-adic, or more generally $S$-arithmetic setting for a finite set of primes $S$. To motivate our results we recall Khintchine’s theorem, a basic result in metric Diophantine approximation. Let $\Psi : {\mathbb{R}}^n \to {\mathbb{R}}_{+} $ be a function satisfying $$\label{defmultapp}
\Psi(a_1, \dots, a_n) \geq \Psi(b_1, \dots, b_n) \text{ if } |a_i| \leq |b_i| \text{ for all } i = 1,\dots, n.$$ Such a function is referred to as a *multivariable approximating function*. Given such a function, define ${\mathcal{W}}_{n}(\Psi)$ to be the set of ${\mathbf{x}}\in {\mathbb{R}}^n$ for which there exist infinitely many ${\mathbf{a}}\in {\mathbb{Z}}^{n}$ 