扭曲丢番图逼近 I：渐近理论
Twisted Diophantine Approximation I: Asymptotic Theory
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- Brandeis University（布兰迪斯大学）
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中文总结 AI 辅助
本文为扭曲丢番图逼近建立精确零一律，通过二进级数收敛性判定，并推广至非齐次情形，恢复已知判据并给出维数结果。
中文摘要 AI 辅助
我们针对具有任意固定实矩阵和满足二进正则性的正非增逼近函数，建立了扭曲丢番图逼近的精确零一律。该判据是某个二进级数的收敛或发散，其加项为Khintchine-Groshev块体积除以齐次格点计数。等价表述是使用相关对角格轨迹的所有逐次极小值，捕捉每个方向上的齐次聚集。我们还获得了非齐次ψ-Lagrange常数的几乎必然零-无穷律、独立的Hausdorff测度判据以及维数结果。在此正则性类中，我们的定理恢复了Kurzweil和Fuchs-Kim判据。在临界指数处，该级数决定了坏逼近平移集合具有零或全Lebesgue测度。
英文摘要
We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous $ψ$-Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.
