2021/06/08 by Elio Joseph, Joseph, Elio
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2106.04313
openalex publication_date 2021/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of ℝn established by W. M. Schmidt in 1967. Let A and B be two subspaces of ℝn of respective dimensions d and e with d+e\leqslant n. The proximity between A and B is measured by t=min(d,e) canonical angles 0\leqslant θ1\leqslant ⋯\leqslant θt\leqslant π/2; we set ψj(A,B)=sinθj. If B is a rational subspace, his complexity is measured by its height H(B)=covol(B∩ℤn). We denote by μn(A\vert e)j the exponent of approximation defined as the upper bound (possibly equal to +∞) of the set of β>0 such that the inequality ψj(A,B)\leqslant H(B)-β holds for infinitely many rational subspaces B of dimension e. We are interested in the minimal value \mathringμn(d\vert e)j taken by μn(A\vert e)j when A ranges through the set of subspaces of dimension d of ℝn such that for all rational subspaces B of dimension e one has dim (A∩ B)