In his works [1], [2] and [3], the author succeeded in establishing several inversion formulas for Radon transform on Euclidean space, DamekRicci space and also on a finite set. The present paper deals with Radon transform R on discrete hyperplanes in the lattice Z n (n ≥ 2) defined by linear diophantine equations. More precisely, we study carefully various natural questions in this context: specific properties of the discrete Radon transform R and its dual R * , inversion formula for R (see Theorem 4.1) and also an appropriate support theorem in the discrete case (see Theorem 5.3).

We study the Radon transform R on the discrete Grassmannian of rank-d affine sublattices of Z n for 0 < d < n, extending and building on previous work of the first-and third-named authors in codimension 1. By analogy with the integral geometry on Grassmannians in R n , various natural questions are treated, such as definition and properties of R and its dual transform R * , function space setting, support theorems and inversion formulas. Mathematics Subject Classification (2000). Primary 44A12; Secondary 05B35, 11D04.

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).

We characterize the image of exponential type functions under the discrete Radon transform R on the lattice Z of the Euclidean space R ( ≥ 2). We also establish the generalization of Volberg's uncertainty principle on Z , which is proved by means of this characterization. The techniques of which we make use essentially in this paper are those of the Diophantine integral geometry as well as the Fourier analysis.