Angular Kronecker Constants for Sets { 1, 2, . . . . . . . ,  n }
(slightly updated June 10, 2015)

• The best known general upper bound is 1/2 - 1/(2 n).
This upper bound applies to any set of n nonzero integers.
The proof can be found in Kronecker Constants for Finite Sets of Integers,
Journal of Fourier Analysis and its Applications,  18(2)  (2012), pages  326-366.
• For the specific sets {1, 2, ...., n} there is a lower bound of 1/2-1/(n+1).
The proof can be found in
Kronecker Constants of Arithmetic Progressions,
with Kathryn E. Hare, Experimental Mathematics 23 (2014), pages 414-422.

 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 ( 159/268, 519/1340 ) ( 2579/4020, 519/1340 ) ( 44/67, 29/67 ) ( 1021/1340, 29/67 )  ( 1021/1340, 121/268 ) ( 159/268, 121/268 ) Click here for a two-dimensional contour plot of this region, with lighter colors corresponding to higher elevations. 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 ( 1661/4494, 863/1498 ) ( 331/749, 863/1498 ) ( 331/749, 887/1498 ) ( 1685/4494, 887/1498 ) Click here for a two-dimensional contour plot of this region, with lighter colors corresponding to higher elevations. lower bound:  0.243051 3/7 8 2/5 1.02857 This occurs at {3/10, 1/2, 3/10, 29/30, 67/120, 9/20, 1/24} {3/10, 1/2, 3/10, 29/30, 67/120, 9/20, 1/16} {3/10, 1/2, 3/10, 137/140, 2157/3920, 131/280, 1/16} {17/40, 3/4, 27/40, 7/15, 3/20, 1/5, 0} {11/35, 3/4, 27/40, 1/2, 6/35, 1/4, 0} {11/35, 3/4, 27/40, 18/35, 6/35, 19/70, 0} {11/35, 3/4, 27/40, 1/2, 6/35, 1/4, 0} among other places (the full exploration of where the extreme occurs remains to be done). 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).

The above-mentioned clever guesses.  This point is fairly hard to approximate but not always the worst case: for integers i in [1,n-2], the i-th coordinate is i/n while the n-1-th coordinate is -1/(2*n).  Here's how this lower bound works for some n:

 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