An Inverse Problem in Number Theory and Geometric Group Theory

  • Melvyn B. NathansonEmail author


This paper describes a new link between combinatorial number theory and geometry. The main result states that A is a finite set of relatively prime positive integers if and only if \(A = (K - K) \cap \mathbf{N}\), where K is a compact set of real numbers such that for every xR there exists yK with xyxymod 1. In one direction, given a finite set A relatively prime positive integers, the proof constructs an appropriate compact set K such that \(A = (K - K) \cap \mathbf{N}\). In the other direction, a strong form of a fundamental result in geometric group theory is applied to prove that (KK) ∩ N is a finite set of relatively prime positive integers if K satisfies the appropriate geometrical conditions. Some related results and open problems are also discussed.


Relatively prime integers Combinatorial number theory Additive number theory Geometric group theory 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsLehman College (CUNY)BronxUSA
  2. 2.CUNY Graduate CenterNew YorkUSA

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