William W. L. Chen

On irregularities of distribution.

§

ON IRREGULARITIES OF DISTRIBUTION II

We study the L w -norm (2 < W < oo) of the discrepancy of a sequence of points in the unit cube relative to similar copies of a given convex polygon. In particular, we prove the rather surprising result that the estimates obtained have the same order of magnitude as the analogous question when the sequence of points is replaced by a set of points.

Geometric discrepancy theory and uniform distribution

A sequence s1, s2, . . . in U = [0, 1) is said to be uniformly distributed if, in the limit, the number of sj falling in any given subinterval is proportional to its length. Equivalently, s1, s2, . . . is uniformly distributed if the sequence of equiweighted atomic probability measures μN (sj) = 1/N , supported by the initial N -segments s1, s2, . . . , sN , converges weakly to Lebesgue measure on U. This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets. Uniform distribution, as an area of study, originated from the remarkable paper of Weyl [Wey16], in which he established the fundamental result known nowadays as the Weyl criterion (see [Cas57, KN74]). This reduces a problem on uniform distribution to a study of related exponential sums, and provides a deeper understanding of certain aspects of Diophantine approximation, especially basic results such as Kronecker’s density theorem. Indeed, careful analysis of the exponential sums that arise often leads to Erdős-Turán type upper bounds, which in turn lead to quantitative statements concerning uniform distribution. Today, the concept of uniform distribution has important applications in a number of branches of mathematics such as number theory (especially Diophantine approximation), combinatorics, ergodic theory, discrete geometry, statistics, numerical analysis, etc. In this chapter, we focus on the geometric aspects of the theory.

Orthogonality and Digit Shifts in the Classical Mean Squares Problem in Irregularities of Point Distribution

Suppose that \( \mathcal{A}_N \) is a distribution of N > 1 points, not necessarily distinct, in the n-dimensional unit cube U n = [0, l) n , where n ≥ 2. We consider the L2-discrepancy

IRREGULARITIES OF POINT DISTRIBUTION RELATIVE TO CONVEX POLYGONS III

\mathcal{L}_2 \left[ {\mathcal{A}_N } \right] = \left( {\int\limits_{U^n } {\left| {\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right]} \right|} ^2 dY} \right)^{1/2} ,

Fourier Techniques in the Theory of Irregularities of Point Distribution

where for every Y = (y1,..., y n) ∈ U n , the local discrepancy \( \mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] \) is given by

On irregularities of distribution III

\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] = \# \left( {\mathcal{A}_N \cap B_Y } \right) - N vol B_Y .