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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Trigonometric approximation and a general form of the Erdős Turán inequality


Authors: Leonardo Colzani, Giacomo Gigante and Giancarlo Travaglini
Journal: Trans. Amer. Math. Soc. 363 (2011), 1101-1123
MSC (2000): Primary 11K38, 42C15
Published electronically: September 20, 2010
MathSciNet review: 2728598
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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\left( x\right) $ and $ B\left( x\right) $ of exponential type $ R$, satisfying $ A(x)\leq\chi_{\Omega}(x)\leq B(x)$ and $ \left\vert B(x)-A(x)\right\vert \leqslant\psi\left( R\operatorname*{dist} \left( x,\partial\Omega\right) \right) $. 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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Additional Information

Leonardo Colzani
Affiliation: Dipartimento di Matematica e Applicazioni, Edificio U5, Università di Milano - Bicocca, via R. Cozzi 53, 20125 Milano, Italia
Email: leonardo.colzani@unimib.it

Giacomo Gigante
Affiliation: Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, viale Marconi 5, 24044 Dalmine, Italia
Email: giacomo.gigante@unibg.it

Giancarlo Travaglini
Affiliation: Dipartimento di Statistica, Edificio U7, Università di Milano - Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italia
Email: giancarlo.travaglini@unimib.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05287-0
PII: S 0002-9947(2010)05287-0
Received by editor(s): February 24, 2009
Received by editor(s) in revised form: December 10, 2009
Published electronically: September 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.