Trigonometric approximation and a general form of the Erdős-Turan inequality

There exists a positive function ψ(t) on t ≥ 0, with fast decay at infinity, such that for every measurable set O in the Euclidean space and R > 0, there exist entire functions A(x) and B (x) of exponential type R, satisfying A(x) ≤ xω(x) ≤ B(x) and |B(x) - A(x)| ≤ψ (Rdist (x,θomega;)). This leads to Erd?os Turán estimates for discrepancy of point set distributions in the multi-dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds. © 2010 American Mathematical Society.