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

• William W. L. Chen
• Maxim M. Skriganov
Conference paper
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

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
$$\mathcal{L}_2 \left[ {\mathcal{A}_N } \right] = \left( {\int\limits_{U^n } {\left| {\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right]} \right|} ^2 dY} \right)^{1/2} ,$$
where for every Y = (y1,..., y n) ∈ U n , the local discrepancy $$\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right]$$ is given by
$$\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] = \# \left( {\mathcal{A}_N \cap B_Y } \right) - N vol B_Y .$$
Here
$$B_Y = \left[ {0,y_1 } \right) \times \ldots \times \left[ {0,y_n } \right) \subseteq U^n$$
is a rectangular box of volume vol By = y1... y n , and #(S) denotes the number of points of a set S, counted with multiplicity.

Keywords

Irregularities of distribution orthogonality digit shift coding theory

11K38

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