Figures & data
Fig. 1 The best current upper and lower bounds for the maximum number of distances in the Euclidean norm (, purple) and the maximum norm (
, green) as a function of the dimension.
![Fig. 1 The best current upper and lower bounds for the maximum number of distances in the Euclidean norm (g¯2d, purple) and the maximum norm (g¯∞d, green) as a function of the dimension.](/cms/asset/7bdaeb3d-b824-4a7e-bf9b-fa7e342e5ba9/uexm_a_2340742_f0001_c.jpg)
Fig. 3 Upper left: plot of in the α plane. The color (black for 1 to yellow for 5) indicates
, that is, the number of distinct shortest distances for N = 9, the given α and Euclidean metric. The region with five distances is too small to be visible and shown in the upper right panel. Lower left and right: Plots of
in the α plane, that is, for N = 11 and the maximum metric.
![Fig. 3 Upper left: plot of g22(α,9) in the α plane. The color (black for 1 to yellow for 5) indicates g22(α,9), that is, the number of distinct shortest distances for N = 9, the given α and Euclidean metric. The region with five distances is too small to be visible and shown in the upper right panel. Lower left and right: Plots of g∞2(α,11) in the α plane, that is, for N = 11 and the maximum metric.](/cms/asset/c43805ac-5d5f-4d06-a8d7-c7b93ae286d6/uexm_a_2340742_f0003_c.jpg)
Fig. 4 Upper left: the Kronecker sequence for , showing five shortest distances in the Euclidean metric,
. Upper right: Formation of the curved triangle and other structure in the upper right panel of . Here, the two components of α are plotted, with
the circular arc defined by
in the Euclidean metric. Lower left: Kronecker sequence for
showing the five shortest distances in the maximum metric,
. Lower right: Formation of the pentagon in the lower right panel of . For
, see Equationequation (9)
(9)
(9) . The boundary of the pentagonal solution set consists of line segments where various combinations of these are equal.
![Fig. 4 Upper left: the Kronecker sequence for α=(0.132,0.38), showing five shortest distances in the Euclidean metric, d2(α,9α)<d2(2α,9α)<d2(3α,9α)<d2(4α,9α)<d2(5α,7α). Upper right: Formation of the curved triangle and other structure in the upper right panel of Figure 3. Here, the two components of α are plotted, with Cm,n the circular arc defined by ⏧mα⏧=⏧nα⏧ in the Euclidean metric. Lower left: Kronecker sequence for α=(0.115,0.314) showing the five shortest distances in the maximum metric, d∞(α,11α)<d∞(2α,11α)<d∞(4α,11α)<d∞(5α,11α)<d∞(6α,5α). Lower right: Formation of the pentagon in the lower right panel of Figure 3. For Nn,j, see Equationequation (9)(9) ⏧nα⏧p={(∑j=1dNn,jp)1/pp<∞max1≤j≤dNn,jp=∞(9) . The boundary of the pentagonal solution set consists of line segments where various combinations of these are equal.](/cms/asset/d453740b-9b13-4e97-929e-a55786939db6/uexm_a_2340742_f0004_c.jpg)
Table 1 Examples where the bound equation (Equation15(15)
(15) ) is tight for the Euclidean norm.
Table 2 Examples where the bound equation (Equation15(15)
(15) ) is tight for the maximum norm.
Table 3 Examples with .