## Abstract

For any three element set of positive integers, \({\{a,b,n \}}\), with *a* < *b* < *n*, *n* sufficiently large and gcd (*a*, *b*)=1, we find the least \({\alpha}\) such that given any real numbers *t*
_{1}, *t*
_{2}, *t*
_{3} there is a real number *x* such that

where \({\langle {\cdot} \rangle}\) denotes the distance to the nearest integer. The number \({\alpha}\) is known as the angular Kronecker constant of \({\{a,b,n \}}\). We also find the least \({\beta}\) such that the same inequality holds with upper bound \({\beta}\) when we consider only approximating *t*
_{1}, *t*
_{2}, *t*
_{3}
\({{\in} \{0,1/2 \}}\), the so-called binary Kronecker constant. The answers are complicated and depend on the congruence of *n* mod (*a* + *b*). Surprisingly, the angular and binary Kronecker constants agree except if \({n{\equiv}a^{2}}\) mod (*a* + *b*).

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

## References

Galindo J., Hernandez S.: The concept of boundedness and the Bohr compactification of a MAP abelian group. Fund. Math.,

**15**, 195–218 (1999)Givens B. N., Kunen K.: Chromatic numbers and Bohr topologies. Top. Appl.,

**131**, 189–202 (2003)C. C. Graham and K. E. Hare, \({\varepsilon}\)-Kronecker and

*I*_{0}sets in abelian groups, I: arithmetic properties of \({\varepsilon}\) Kronecker sets,*Math. Proc. Camb. Philos. Soc*.,**140**(2006), 475–489.C. C. Graham, K. E. Hare and T. W. Korner, \({\varepsilon}\)-Kronecker and

*I*_{0}sets in abelian groups, II: sparseness of products of \({\varepsilon}\)-Kronecker sets,*Math. Proc. Camb. Philos. Soc*.,**140**(2006), 491–508.Hare K. E., Ramsey L. T.: Kronecker constants for finite subsets of integers. J. Fourier Anal. Appl.,

**18**, 326–366 (2012)Hare K. E., Ramsey L. T.: Exact Kronecker constants of Hadamard sets. Colloq. Math.,

**130**, 39–49 (2013)K. E. Hare and L. T. Ramsey, Kronecker constants of arithmetic progressions,

*Experimental Math*. (2014), to appear.Kunen K., Rudin W.: Lacunarity and the Bohr topology, Math. Proc. Camb. Philos. Soc.,

**126**, 117–137 (1999)G. Nemhauser and L. Wolsey,

*Integer and Combinatorial Optimization*, Wiley and Sons (New York, 1988).Varapoulos N.: Tensor algebras and harmonic analysis, Acta Math.,

**119**, 51–112 (1968)

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

This work was partially supported by NSERC through application number 44597, and the Edinburgh Math Soc. The authors would like to thank St. Andrews University for their hospitality when some of this research was done.

## Rights and permissions

## About this article

### Cite this article

Hare, K.E., Ramsey, L.T. Exact Kronecker constants of three element sets.
*Acta Math. Hungar.* **146**, 306–331 (2015). https://doi.org/10.1007/s10474-015-0529-2

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10474-015-0529-2