Trigonometric approximation and a general form of the Erd\H{o}s Tur\'{a}n inequality

TL;DR

This paper develops a general framework for approximating characteristic functions using entire functions with controlled error, leading to new discrepancy estimates in multi-dimensional torus and manifold settings.

Contribution

It introduces a novel approximation method using trigonometric and eigenfunction techniques to derive generalized Erdős–Turán inequalities for discrepancy analysis.

Findings

Existence of entire functions approximating characteristic functions with controlled error.

Derivation of discrepancy bounds in multi-dimensional torus.

Extension of results to eigenfunctions on compact manifolds.

Abstract

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.