Trigonometric approximation and a general form of the Erdős Turán inequality

There exists a positive function $ψ(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$Ω${in the Euclidean space and}$R>0${, there exist entire functions}$A(x)${and}$B(x)${of exponential type}$R${, satisfying\}$A(x)\leq χ_Ω(x)\leq B(x)${and}$| B(x)-A(x)| \leqslantψ(R\operatorname*{dist}(x,\partialΩ))$. 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.

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