Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines - Archive ouverte HAL
Article Dans Une Revue Annali della Scuola Normale Superiore di Pisa, Classe di Scienze Année : 2022

Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines

Dmitry Kleinbock
  • Fonction : Auteur
Nimish Shah
  • Fonction : Auteur
Pengyu Yang
  • Fonction : Auteur

Résumé

Let X = SL3(R)/ SL3(Z), and gt = diag(e 2t , e −t , e −t). Let ν denote the push-forward of the normalized Lebesgue measure on a straight line segment in the expanding horosphere of {gt}t>0 by the map h → h SL3(Z), SL3(R) → X. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families in X: (1) gt-translates of ν as t → ∞. (2) averages of gt-translates of ν over t ∈ [0, T ] as T → ∞. (3) gt i-translates of ν for some ti → ∞. We apply this dynamical result to show that Lebesgue-almost every point on the planar line y = ax + b is not Dirichlet-improvable if and only if (a, b) / ∈ Q 2 .
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Dates et versions

hal-03853345 , version 1 (15-11-2022)

Identifiants

  • HAL Id : hal-03853345 , version 1

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Dmitry Kleinbock, Nicolas de Saxcé, Nimish Shah, Pengyu Yang. Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, In press. ⟨hal-03853345⟩
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