Winning and nullity of inhomogeneous bad

Abstract: 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].