Additive Number Theory pp 249-258 | Cite as

# An Inverse Problem in Number Theory and Geometric Group Theory

## Summary

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 *x* ∈ **R** there exists *y* ∈ *K* with *x* ≡ *yx* ≡ *y*mod 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 (*K* − *K*) ∩ **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.

## Keywords

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

- 1.P. de la Harpe,
*Topics in Geometric Group Theory*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.zbMATHGoogle Scholar - 2.V. A. Efremovič,
*The proximity geometry of Riemannian manifolds*, Uspekhi Math. Nauk**8**(1953), 189.Google Scholar - 3.J. Milnor,
*A note on curvature and fundamental group*, J. Diff. Geom.**2**(1968), 1–7.MathSciNetzbMATHGoogle Scholar - 4.M. B. Nathanson,
*Phase transitions in infinitely generated groups, and related problems in additive number theory*, arXiv: 0811.3990, 2008. Integers, to appear.Google Scholar - 5.M. B. Nathanson,
*Nets in groups, minimum length g-adic representations, and minimal additive complements*, arXiv: 0812.0560, 2008.Google Scholar - 6.M. B. Nathanson,
*Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory*, arXiv: 0902.3254, 2009.Google Scholar - 7.A. S. Švarc,
*A volume invariant of coverings*, Dokl. Akad. Nauk SSSR (N.S.)**105**(1955), 32–34.Google Scholar