Difference sets and inverting the difference operator

Abstract

For a setA of non-negative numbers, letD(A) (the difference set ofA) be the set of nonnegative differences of elements ofA, and letD k be thek-fold iteration ofD. We show that for everyk, almost every set of non-negative integers containing 0 arises asD k (A) for someA. We also give sufficient conditions for a setA to be the unique setX such that 0∈X andD k (X)=D k (A). We show that for eachm there is a setA such thatD(X)=D(A) has exactly 2m solutionsX with 0∈X.

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Lee A. Rubel died March 25, 1995. He is very much missed by his coauthors.

This work was supported by grants DMS 92-02833 and DMS 91-23478 from the National Science Foundation. The first author acknowledges the support of the Hungarian National Science Foundation under grants, OTKA 4269, and OTKA 016389, and the National Security Agency (grant No. MDA904-95-H-1045).

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Füredi, Z., Jockusch, C.G. & Rubel, L.A. Difference sets and inverting the difference operator. Combinatorica 16, 87–106 (1996). https://doi.org/10.1007/BF01300128

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Mathematics Subject Classification (1991)

  • 11 B 05
  • 05 C 65