Winning of inhomogeneous bad for curves

TL;DR

This paper proves that weighted inhomogeneous badly approximable vectors on non-degenerate analytic curves possess the absolute winning property, extending previous results and contributing to the inhomogeneous Schmidt's conjecture across dimensions.

Contribution

It establishes the absolute winning property for inhomogeneous badly approximable vectors on curves, extending prior work and solving an open question in the field.

Findings

Proves absolute winning property for inhomogeneous vectors on curves

Extends previous results to inhomogeneous setting and higher dimensions

Contributes to the proof of inhomogeneous Schmidt's conjecture

Abstract

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.