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.
Categories: math.NT math.CA