  There exists a positive function $\psi(t)${on}$t\geq0${, with fast decay at
infinity, such that for every measurable set}$\Omega${in the Euclidean space
and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential
type}$R${, satisfying\}$A(x)\leq \chi_{\Omega}(x)\leq B(x)${and}$| B(x)-A(x)|
\leqslant\psi(R\operatorname*{dist}(x,\partial\Omega)) $. This leads to
Erd\H{o}s Tur\'{a}n 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.
