Tom Ramsey
Emeritus Professor

Computing Angular Kronecker Constants (joint work with Kathryn Hare)

• Some partial data about arithmetic progressions, { 1, 2, 3, .  .  . , n }  (updated slightly, June 10, 2015)
• A text file of angular constants for three element sets { a, b, c } with 0 < a < b < c < 51 and the gcd(a,b,c) = 1. (posted June 23, 2008)
  Comment added 4/8/2014:  The angular constant for {1, 4, 18} is 5/38.  Note that {1, 4, 18} is lacunary with ratio q=4 but that 5/38>1/(2q)=1/8.  This phenomenon is not unusual.  The text file includes 8 other examples, three with q=5 and five with q=6.
• A text file of angular constants for relatively prime, four element subsets from [1 ... 20] , with one additional 4 element set (posted October 1, 2008)

Papers about Kronecker Constants (updated Jan. 7, 2016)
co-authored with Kathryn Hare (University of Waterloo)

• The Relationship between Epsilon-Kronecker and Sidon Sets, pre-print.  This paper uses a criterion due to Pisier to prove that Epsilon-Kronecker implies Sidon, when Epsilon<2.  It also generalizes the result to weaker notions of Epsilon-Kronecker.  The paper will appear in the Canadian Mathematical Bulletin (http://cms.math.ca/10.4153/CMB-2016-002-3) with document identifier 10.4153/CMB-2016-002-3
  
  A preprint is available at http://arxiv.org/abs/1506.07389
  .
• Exact Kronecker Constants of Three Elements Sets, Acta Mathematica Hungarica, Volume 146, Issue 2, pages 306-331.   DOI:  10.1007/s10474-015-0529-2
  Available on-line at http://link.springer.com/article/10.1007/s10474-015-0529-2
  Also see a preprint at http://arxiv.org/abs/1503.09071
  
  This paper gives exact formulas for positive integers sets {a, b, n} with gcd(a,b)=1 and n relatively large.  The formula depends on the residue of n mod ( a + b ).  For all but one of the residue classes, the formula agrees with the binary Kronecker constant.
  
  
• Kronecker Constants of Arithmetic Progressions, with Kathryn E. Hare, Experimental Mathematics 23 (2014), pages 414-422.
  
  This paper provides a lower bound of 1/2 - 1/d for the angular Kronecker constant of the progressions 1, 2, ...., d-1.  This lower bound equals the worst approximation error when approximating all vectors with coordinates 0 or 1/2 (sometimes called the binary angular Kronecker constant).   The lower bound contrasts nicely with the upper bound of 1/2 -1/(2d-2) which Kathryn Hare and I proved earlier for all sets of d-1 positive integers.
  
  
• Exact Kronecker Constants of Hadamard Sets,  with Kathryn E. Hare, Colloquium Mathematicum 130 (2013), pages 39-49.
  For integers m>1 and d >0, the exact constant is provided for the set consisting of 1, m, ..., m^d-1.  
  This is then used to give the exact constant for the infinite sequence 1, m, m², ....  
  The theory extends to some classes of sets of the form 1, m₁, m₁m₂, m₁m₂m₃, ....
  
  
• Kronecker Constants for Finite Sets of Integers (with the permission of the publisher, here is a personally generated PDF of the paper), to appear in the Journal of Fourier Analysis and Applications.  The final publication is available at  www.springerlink.com
  
  
• Upper and Lower Bounds for Kronecker Constants of Three-Element Sets of Integers,  arXiv:1108.3802.  This paper is a supplement to Kronecker Constants for Finite Sets of Integers

A New Paper in Statistics (July, 2011)

• Maximum-Likelihood Non-Decreasing Response Estimates.  It has been posted to http://arxiv.org/abs/1107.1025v1

Two recent papers co-authored with Colin C. Graham:

• Planar Sidonicity and quasi-independence for multiplicative subgroups of the roots of unity. Pacific J. Math. 225 (2006), 325-360.
• Permutations and extensions of planar quasi-independent subgroups of the roots of unity. In preparation (2006).
• Some background papers about Sidon sets (published in Colloquium Mathematicum):
  
  1. Bohr Cluster points of Sidon sets.
  2. Comparisons of Sidon and I₀ Sets.
  3. Proportions of Sidon Sets Are I₀ Sets.
  4. On Partitioning Sidon Sets with Quasi-Independent Sets
     (with K. T. Harrison).

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