2021/06/11 by Elio Joseph, Joseph, Elio
Computer Science · Mathematics · #Digital Image Processing Techniques #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2106.06386
openalex publication_date 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper uses W. M. Schmidt's idea formulated in 1967 to generalise the classical theory of Diophantine approximation to subspaces of ℝn. Given two subspaces of ℝn A and B 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 for infinitely many rational subspaces B of dimension e, the inequality ψj(A,B)\leqslant H(B)-β holds. 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)