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: 1/2-1/(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 half-domain 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 2-dimensional interior) [ 319/470, 586/705 ] x [ 243/470, 123/235 ] x { 201/235 } x { 3/235 } Click here for a two-dimensional 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 2-dimensional interior) whose 6 corners going counterclockwise and starting from the lower left corner are ( x1, 21/67, x3, 97/134, 1/268 ) with ( x1, x3 ) 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 2-dimensional
interior) whose 4 corners going counterclockwise starting from the
lower left are ( x1, 583/749, x3,
512/749, 599/749, 2/749 ) with ( x1, x3
) 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 3-dimensional 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 8-th 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 ((2-k2))/2).
If the set under consideration has d integers, there is a discrete subgroup K of Rd-1 for which a is the farthest distance of any point p in Rd-1 from K. One can prove that such a farthest point exists and occurs in the "half domain" [0,1)d-2 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(pi-pi/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 | logn(1/2-lower 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 |