# Jill Pipher : Geometric discrepancy theory: directional discrepancy in 2-D

Discrepancy theory originated with some apparently simple questions about
sequences of numbers. The discrepancy of an infinite sequence is a
quantitative measure of how far it is from being uniformly distributed.
Precisely, an infinite sequence { a1,a2, ...} is said to be uniformly
distributed in [0, 1] if

lim_{n\to\infty} (1/n|{a1, a2,...an} intersect [s,t]|) = t-s.

If a sequence {ak} is uniformly distributed, then it is also the case
that for all (Riemann) integrable functions f on [0, 1],

lim_{n\to\infty} (1/n\sum_{k=1}^n f(ak))=\int_0^1 f(x)dx.

Thus, uniformly distributed sequences provide good numerical schemes
for approximating integrals. For example, if alpha is any irrational
number in [0, 1], then the fractional part {alphak}:=ak is uniformly
distributed. Classical Fourier analysis enters here, in the form of
Weyl's criterion.
The discrepancy of a sequence with respect to its first n entries is

D({ak},n) := sup_{s

See PDF.

**Category**: Applied Math and Analysis**Duration**: 01:34:52**Date**: October 3, 2011 at 4:25 PM**Views**: 101-
**Tags:**seminar, Applied Math And Analysis Seminar

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