Interval discrepancy ($D_N(I)={\sum }_{j=1}^N{1}_I(u_j)-N|I|$)

Interval discrepancy measures the difference between empirical counts and the expected count for normalized Lebesgue measure. Convolving an enlarged interval indicator function with a narrow unit-integral indicator function gives a continuous majorant. Its Fourier coefficients decay as $O(1/(\delta k^2))$, converting the count into a constant term, finite exponential sums and a controlled absolute tail.