Birkhoff generic points on curves in horospheres

• Omri Nisan Solan ^[1] ; Andreas Wieser ^[2]
  1. [1] Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Hebrew University, Jerusalem, Israel
  2. [2] Institute for Advanced Study, 1 Einstein Drive, Princeton , USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01096-9
• Enlaces
  • Texto completo
• Resumen

  • Let {at : t ∈ R} < SLd (R) be a diagonalizable subgroup whose expanding horospherical subgroup U < SLd (R) is abelian. By the Birkhoff ergodic theorem, for any x ∈ SLd (R)/ SLd (Z) and for almost every point u ∈ U the point ux is Birkhoff generic for at when t → ∞. We prove that the same is true when U is replaced by any non-degenerate analytic curve in U. This Birkhoff genericity result has various applications in Diophantine approximation. For instance, we obtain density estimates for Dirichlet improvability along typical points on a curve in Euclidean space. Other applications address approximations by algebraic numbers and best approximations (in the sense of Lagarias).

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