Trigonometric approximation and a general form of the Erdős Turán inequality

Abstract:

There exists a positive function $\psi (t)$ on $t\geq 0$, with fast decay at infinity, such that for every measurable set $\Omega$ in the Euclidean space and $R>0$, there exist entire functions $A\left ( x\right )$ and $B\left ( x\right )$ of exponential type $R$, satisfying $A(x)\leq \chi _{\Omega }(x)\leq B(x)$ and $\left \vert B(x)-A(x)\right \vert \leqslant \psi \left ( R\operatorname *{dist} \left ( x,\partial \Omega \right ) \right )$. This leads to Erdős 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.