n  Angular Kronecker Constant (AKC)  Ratio of AKC to Exp. Math. Lower Bound  Where Achieved within the associated "half domain"  Previously Published Estimate (Graham and Hare)  JFAA Published
Upper Bound: 1/21/(2n) 
1  0  0  0  
2  1/6  1  1/4  
3  1/4  1  The closed line segement from (1/3,1/12) to (1/3,1/6). For a further description of this, click here.  lower bound: 0.04488  1/3 
4  3/10  1  The closed line segment from (3/8, 11/60, 1/40) to (3/8, 19/60, 9/40), whose center is (3/8, 1/4,1/8). The segment from (3/8, 1/4, 1/8) to (3/8, 19/60, 9/40) is outside the "half domain" and is within a conjugate halfdomain that is obtained by reflecting the canonical half domain through the plane z = 1/8 which forms part of the boundary of the canonical half domain.  lower bound: 0.139326  3/8 
5  16/47  1.10213  The closed rectangle (with its 2dimensional interior) [ 319/470, 586/705 ] x [ 243/470, 123/235 ] x { 201/235 } x { 3/235 } Click here for a twodimensional contour plot of this region, with lighter colors corresponding to higher elevations. 
lower bound: 0.189333  2/5 
6  49/134  1.02388  The closed polygon (with its 2dimensional interior) whose 6 corners going counterclockwise and starting from the lower left corner are ( x_{1}, 21/67, x_{3}, 97/134, 1/268 ) with ( x_{1}, x_{3} ) being

lower bound: 0.220945  5/12 
7  41/107  1.02181  The set where this occurs is conjectured to
have affine dimension 4 and also conjectured to be the convex hull of
12 points. Part of this set is the following closed trapezoid
(with its 2dimensional
interior) whose 4 corners going counterclockwise starting from the
lower left are ( x_{1}, 583/749, x_{3},
512/749, 599/749, 2/749 ) with ( x_{1}, x_{3}
) being

lower bound: 0.243051  3/7 
8  2/5  1.02857  This occurs at

lower bound: 0.259551  7/16 
9  12/29  1.03488  An exhaustive search
found a 3dimensional box rectangular on which the maximum occurred.
One corner of the box is {236/261, 401/522, 19/174, 325/522, 31/87, 85/87, 997/2088, 1/261} To obtain the other corners move 1/174 in the first coordinate, 1/174 in the second coordinate, and 357/4872 in the third coordinate. The search space had been subdivided into rectangular parallelotopes that were 1/6 wide in each coordinate, except for the 8th which was always the interval [0, 1/18]. Only two of these contained points at which the maximum occurred, and their union is [5/6,1] x [2/3, 5/6] x [0, 1/3] x [1/2, 2/3] x [1/3, 1/2] x [5/6, 1] x [1/3, 1/2] x [0, 1/18] 
lower bound: 0.27244  4/9 
10  to be cont'd  to be cont'd  The Exp. Math. lower bound is 9/22. There are examples of target vectors that give a higher answer. In particular, the constant for n=9 is 12/29>9/22.  lower bound: 0.282853  9/20 
11  not done yet  not done yet  The Exp. Math. lower bound is 5/12.  lower bound: 0.291484 Also, 0.399942 is a lower bound observed as a special case 
5/11 
The angular Kronecker constant is a constant a such that exp(2 pi i a)  1 is the Kronecker constant. The inverse relation for the angular constant in terms of the Kronecker constant is that a = arccos ((2k^{2}))/2).
If the set under consideration has d integers, there is a discrete subgroup K of R^{d1} for which a is the farthest distance of any point p in R^{d1} from K. One can prove that such a farthest point exists and occurs in the "half domain" [0,1)^{d2} x [0,1/(2*n)].
Graham and Hare have published previously the following lower bound for Kronecker constants: If exp(pi/delta) < n, then k{1,2...,n}>= 1  exp(i(pi  delta)). By continuity, we have k(1, 2, ..., n) >=  1  exp(i(pipi/ln(n))  which gives a lower bound of (1  1/ln(n) )/2 for the angular Kronecker constant. By a separate calculation made when n = 11, a lower bound on k was published that is 1.902 which gives a lower bound of 0.399942 for the angular constant. Notice that the Graham and Hare general lower bound converges to 2 as n goes to infinity (and hence the angular constant converges to 1/2).n  clever point lower bound  relative error  log_{n}(1/2lower bound) 
3  1/4  0  1.26186 
4  3/10  0  1.16096 
5  1/3  1/48 (0.0208)  1.11328 
6  5/14  8/335 (0.0239)  1.08063 
7  3/8  7/328 (0.0213)  1.06862 
8  7/18  unknown  1.05664 
9  2/5  unknown  1.04795 
10  
11  
12 