Erdős-Turán inequality

The Erdős-Turán inequality bounds how unevenly finitely many points fill the half-open interval $[0,1)$ by measuring finitely many Fourier modes. For $x_1,\dots ,x_N\in [0,1)$ define their normalized star discrepancy byThere is a universal constant $C$ such that, for every positive integer $H$,The finite-point form follows from the corresponding probability measure inequality by placing mass $1/N$ at each point of the unit circle represented by $x_n$.^[ref]

The cutoff $H$ trades an approximation error against the measured exponential sums. If each fixed nonzero Fourier mode tends to zero as $N$ grows, take the limit superior first with $H$ fixed, then let $H$ grow. This gives $D_N^{\ast }\to 0$, the quantitative route to equidistribution.

For the grid $x_n=(n-1)/N$ with $N\ge 2$, every displayed exponential sum vanishes for $1\le h<N$ by the finite geometric series. Choosing $H=N-1$ yields $D_N^{\ast }\le C/(N-1)$; direct counting gives the sharper exact value $1/N$.