A decomposition theorem for homogeneous sets with respect to diamond probes

An unknown planar discrete set of points A can be inspected by means of a probe P of generic shape that moves around it, and reveals, for each position, the number of its elements as a magnifying glass. All the data collected during this process can be naturally arranged in an integer matrix that we call the scan of the starting set A w.r.t. the probe P. In [10], Nivat conjectured that a discrete set whose scan w.r.t. an exact probe is k-homogeneous, shows a strong periodical behavior, and it can be decomposed into smaller 1-homogeneous subsets. In this paper, we prove this conjecture to be true when the probe is a diamond, and then we extend this result to exact polyominoes that can regarded as balls in a generalized L-1 norm of Z(2). Then we provide experimental evidence that the conjecture holds for each exact polyomino of small dimension, using the mathematical software Sage [13]. Finally, we give some hints to solve the related reconstruction problem.