ISRAEL JOURNAL OF MATHEMATICS TBD (2017), 1–25 DOI: 10.1007/s11856-017-1550-7 ERROR DIFFUSION ON ACUTE SIMPLICES: INVARIANT TILES BY Roy Adler, Tomasz Nowicki, Grzegorz ´ Swirszcz, Charles Tresser and Shmuel Winograd IBM T. J. Watson Research Center 33-210 1101 Kitchawan Road Rte 134, PO Box 218, Yorktown Heights, NY 10598, USA e-mail: rla@us.ibm.com, tnowicki@us.ibm.com, swirszcz@us.ibm.com, charlestresser@yahoo.com, swino@us.ibm.com ABSTRACT We study the absorbing invariant set of a dynamical system defined by a map derived from Error Diffusion, a greedy online approximation algo- rithm that minimizes the (Euclidean) norm of the cumulated error. This algorithm assigns a sequence of outputs, each a vertex of some polytope, to any sequence of inputs in that polytope. Here, the polytope is assumed to be a simplex that is acute, meaning that the pairs of distinct external normal vectors to the codimension-one faces form obtuse angles. The in- put is assumed to be constant. The map is a system which consists of piecewise translations acting on the partition of an affine space into the Vorono¨ ı regions defined (once tie-breaking is resolved) by the vertices of the polytope. The translation vector in each partition piece is the diffe- rence between the input modified by adding the cumulated error and the corresponding vertex. When the polytope is a simplex such piecewise translation projects onto a translation of a torus. We prove that if the projected translation is ergodic, then there is an invariant absorbing set of the piecewise trans- lation which is a fundamental set for the lattice generated by the simplex and which is a limit set of any bounded set with nonempty interior. Received August 21, 2013 and in revised form November 11, 2015 1