Functional tilings and the Coven-Meyerowitz tiling conditions

Abstract

Coven and Meyerowitz [1] formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group ZM. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions f,g, such that f(0) = g(0) = 1, f ∗g = 1ZM is a functional tiling, and f,g satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.