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Volume 9, Issue 3


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On the Beauty of Uniform Distribution Modulo One
Karl Entacher


The theory of uniform distribution modulo one was developed extensively within and among several mathematical disciplines and numerous applications, mostly in the fields of Monte Carlo and quasi-Monte Carlo methods, which include areas like numerical integration, random number generation, stochastic simulation, and approximation theory.

The central goals of this theory are the assessment of equidistribution and the construction of well-distributed point sets and sequences in various mathematical spaces.

The following sections contain several supporting ideas for introducing the theory of uniform distribution in education, offer additional information for researchers, and supplement the theory with impressive images. We start with some elementary examples. Section 2 treats discrepancy, which is the classical measure of uniform distribution. In Section 3 we use a special graphical presentation of local discrepancy, showing the beauty of uniform distribution. Section 4 considers further examples of point sets and the graphical visualization of the quality of their distribution.

Classical and recent concepts of the theory and further references are discussed in [1, 2]. For further information on quasi-Monte Carlo methods and their applications, see [3, 4, 5]. For efficient Mathematica implementations of "quasi-random numbers," see QR Stream [6]. Note that only small point sets are used for our illustrations; the number of points used in practice is substantially larger.

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