Integral geometry on discrete Grassmannians 𝔾(d, n) associated to ℤ n

Abstract

The purpose of this paper is to extend carefully the discrete Radon transform, studied in [A. Abouelaz and A. Ihsane, Diophantine integral geometry, Mediterr. J. Math. 5(1) (2008), pp. 77–99], to the Radon transform R on the discrete Grassmannian 𝔾(d, n) (with n≥3 and 1≤d<n−1) consisting of all discrete d-planes in the lattice ℤ n defined by systems of linear diophantine equations. By analogy with the integral geometry on Grassmann manifolds and projective spaces, which was developed by many authors, this study deals with various natural questions in this context: specific properties of the discrete Radon d-plane transform R and its dual R*, inversion formula for R (see Theorem 5.1) and also an appropriate support theorem for this Radon transform (see Theorem 6.3).