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arXiv · 2605.19582

The inhomogeneous Khintchine Theorem in dimension two

Demi Allen·Manuel Hauke-Treuer·Felipe A. Ramírez
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Abstract

We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine--Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart.

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Demi Allen, Manuel Hauke-Treuer, Felipe A. Ramírez. 2026-05-19. The inhomogeneous Khintchine Theorem in dimension two. https://arxiv.org/abs/2605.19582

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