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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 $x_1,\dots ,x_N\in [0,1)$ 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 $1/N$.

Ohad N. Feldheim coauthored a proof of the Erdős-Turán inequality and an error estimate for Wigner semicircle distribution with Sasha Sodin. The discrepancy bound controls deviations of a measure on the unit circle through finitely many Fourier coefficients. It converts harmonic information into a quantitative statement about spatial distribution.^[ref]

Proposition 1 states the Erdős-Turán inequality for a probability measure on the unit circle, with a universal constant and normalized arc length. The finite-point version follows by taking an empirical measure.

Sasha Sodin studied the Erdős-Turán inequality and quantitative errors in Wigner semicircle distribution with Ohad N. Feldheim. Their work connects estimates on Fourier or polynomial moments with discrepancies between probability measures. Such bounds make convergence of global spectral distributions quantitative, rather than merely asserting a limiting law.^[ref]