 Star discrepancy

For points $x_1,\dots ,x_N$ in $[0,1{)}^d$, star discrepancy compares their empirical frequencies in anchored boxes with Lebesgue measure:The half-open box convention makes boundary membership precise. Small star discrepancy means that every such box contains nearly its uniform share of points; it does not mean the points are independent random samples.

In one dimension, take midpoint points $x_i=(i-1/2)/N$. Between consecutive points, the empirical count is constant while the interval length increases linearly. Just before or after a point the largest difference is $1/(2N)$, so $D_N^{\ast }=1/(2N)$. Repeating a single point instead leaves large gaps and large discrepancy. The definition directly controls the integration error for indicators of anchored boxes and motivates deterministic sampling for numerical integration.