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

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 Author Colzani, Leonardo ♦ Gigante, Giacomo ♦ Travaglini, Giancarlo Source arXiv.org Content type Text File Format PDF Date of Submission 2010-01-06 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Number Theory ♦ Mathematics - Classical Analysis and ODEs ♦ 11K38 ♦ 42C15 ♦ math 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. Educational Use Research Learning Resource Type Article