= Erdős-Turán inequality
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The <Erdős-Turán inequality> bounds how unevenly finitely many points fill the <half-open interval> $[0,1)$ by measuring finitely many <Fourier modes>. For $x_1,\ldots,x_N\in[0,1)$ define their normalized <star discrepancy> by
$$
D_N^*=\sup_{0\leq t\leq1}\left|\frac1N\#\{n:x_n<t\}-t\right|.
$$
There is a universal constant $C$ such that, for every <positive integer> $H$,
$$
D_N^*\leq C\left(\frac1H+\sum_{h=1}^H\frac1h\left|\frac1N\sum_{n=1}^N e^{2\pi ihx_n}\right|\right).
$$
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 $x_n$.<One more proof of the Erdős-Turán inequality, and an error estimate in Wigner's law by Ohad N. Feldheim and Sasha Sodin>[Proposition 1, equation (1), printed p. 1; rescale the circle coordinate by $2\pi$ and use the empirical probability measure]{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 $D_N^*\to0$, the quantitative route to <equidistribution>.
For the grid $x_n=(n-1)/N$ with $N\geq2$, every displayed <exponential sum> vanishes for $1\leq h<N$ by the finite <geometric series>. Choosing $H=N-1$ yields $D_N^*\leq C/(N-1)$; direct counting gives the sharper exact value $1/N$.
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