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On the Beauty of Uniform Distribution Modulo One

Visualizations

In this section, we visualize the structure of local discrepancy for our point sets , for all discrete intervals with resolution . We start with a visualization in dimension two. Therefore, we use DiscreteDiscrepancy[ ] and calculate the list of values for all such two-dimensional intervals, which are represented by their endpoints as described earlier. The following function simply uses RasterArray[ ] to color the list of local discrepancies. Small values of are colored in dark blue. The larger the values become, the brighter the coloring. The brightest squares show where the maximum (and therefore the discrete discrepancy) is attained. The explicit value is shown above each plot.

Figure 4 shows Visualize2D[p,m] for and . Compared to Figure 2, these graphics reflect the change of distribution in a more impressive way.

Figure 4. Visualization of local discrepancy for , and .

Figure 5 also includes the points within the graphic for the larger case and . A comparison of the first graphic in Figures 4 and 5 shows certain self-similarities. The reader is also asked to visualize the cases .

Figure 5. Visualization of local discrepancy of for .

In Figure 6, the distribution behavior of the point sets is visualized in three dimensions as well. The function Visualize3D[p,m] generates a Graphics3D object containing cuboids with heights corresponding to . Figures 6 and 7 show the cases , , and . Two of the plots in Figure 4 are projections into the x-y plane of Figures 6 and 7.

Figure 6. Three-dimensional visualization of local discrepancy of for .

Figure 7. Three-dimensional visualization of local discrepancy of for .