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  • 1 Eötvös Loránd University Department of Algebra and Number Theory Pázmány Péter sétány 1/c H-1117 Budapest Hungary
  • 2 Eötvös Loránd University Department of Analysis Pázmány Péter sétány 1/c H-1117 Budapest Hungary
  • 3 Computer and Automation Research Institute Kende u. 13–17 H-1111 Budapest Hungary
  • 4 Hungarian Academy of Sciences Alfréd Rényi Institute of Mathematics P.O. Box 127 H-1364 Budapest Hungary
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Abstract  

We study those functions that can be written as a finite sum of periodic integer valued functions. On ℤ we give three different characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a ℤ → ℤ function implies the existence of an integer valued periodic decomposition with the same periods. This result depends on the representation of the greatest common divisor of certain polynomials with integer coefficients as a linear combination of the given polynomials where the coefficients also belong to ℤ[x]. We give an example of an ℤ → {0, 1} function that has a bounded real valued periodic decomposition but does not have a bounded integer valued periodic decomposition with the same periods. It follows that the class of bounded ℤ → ℤ functions has the decomposition property as opposed to the class of bounded ℝ → ℤ functions. If the periods are pairwise commensurable or not prescribed, then we get more general results.