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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 consider the Radon transform on the (flat) torus T n = R n /Z n defined by integrating a function over all closed geodesics. We prove an inversion formula for this transform and we give a characterization of the image of the space of smooth functions on T n .Mathematics Subject Classification (2010). Primary 53C65, 44A12.

We define and study the d-plane Radon transform, namely R, on the n-dimensional (flat) torus. The transformation R is obtained by integrating a suitable function f over all d-dimensional geodesics (d-planes in the torus). We specially establish an explicit inversion formula of R and we give a characterization of the image, under the d-plane Radon transform, of the space of smooth functions on the torus.

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