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arXiv · 2210.10504

Diophantine approximation with constraints

Jérémy Champagne·Damien Roy
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Abstract

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $\mathbb{R}^n$. Assuming that the point of $\mathbb{R}^n$ that we are approximating has linearly independent coordinates over $\mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.

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Jérémy Champagne, Damien Roy. 2022-10-19. Diophantine approximation with constraints. https://arxiv.org/abs/2210.10504

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