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Diophantine approximation with constraints

Volume 207 / 2023

Jérémy Champagne, Damien Roy Acta Arithmetica 207 (2023), 57-99 MSC: Primary 11J13; Secondary 11H50, 11J25. DOI: 10.4064/aa221031-8-12 Published online: 2 February 2023


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.


  • Jérémy ChampagneDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, Ontario N2L 3G1, Canada
  • Damien RoyDépartement de Mathématiques
    Université d’Ottawa
    Ottawa, Ontario K1N 6N5, Canada

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