Periodicity of non-central integral arrangements modulo positive   integers
  An integral coefficient matrix determines an integral arrangement of hyperplanes in R^m. After modulo q reduction, the same matrix determines an arrangement A_q of "hyperplanes" in Z^m. In the special case of central arrangements, Kamiya, Takemura and Terao [J. Algebraic Combin., to appear] showed that the cardinality of the complement of A_q in Z_q^m is a quasi-polynomial in q. Moreover, they proved in the central case that the intersection lattice of A_q is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement B_m^{[0,a]} of Athanasiadis [J. Algebraic Combin. Vol.10 (1999), 207-225] to illustrate our results. 
Hidehiko Kamiya and Akimichi Takemura and Hiroaki Terao