Inégalité d'Erdős–Turán

En mathématiques, l'inégalité d'Erdős-Turán majore la distance entre une mesure de probabilité sur le cercle et la mesure de Lebesgue, en termes de coefficients de Fourier. Elle fut prouvé par Paul Erdős et Pál Turán en 1948^[1]^,^[2].

Soit μ une mesure de probabilité sur le cercle unité R/Z. L'inégalité d'Erdős-Turán énonce que, pour tout nombre naturel n,

\sup _{A}\left|\mu (A)-\lambda (A)\right|\leq C\left({\frac {1}{n}}+\sum _{k=1}^{n}{\frac {|{\hat {\mu }}(k)|}{k}}\right),

où le supremum porte sur l'ensemble des arcs A ⊂ R/Z du cercle unité, λ représente la mesure de Lebesgue,

{\hat {\mu }}(k)=\int \exp(2\pi ik\theta )\,d\mu (\theta )

sont les coefficients de Fourier de μ, et C > 0 est une constante numérique.

Application à la divergence

Soit s₁, s₂, s₃... ∈ R une suite. L'inégalité d'Erdős-Turán appliquée à la mesure

\mu _{m}(S)={\frac {1}{m}}\#\{1\leq j\leq m\,|\,s_{j}\,\mathrm {mod} \,1\in S\},\quad S\subset [0,1),

donne la borne suivante pour la discrépence :

{\begin{aligned}D(m)&\left(=\sup _{0\leq a\leq b\leq 1}{\Big |}m^{-1}\#\{1\leq j\leq m\,|\,a\leq s_{j}\,\mathrm {mod} \,1\leq b\}-(b-a){\Big |}\right)\\[8pt]&\leq C\left({\frac {1}{n}}+{\frac {1}{m}}\sum _{k=1}^{n}{\frac {1}{k}}\left|\sum _{j=1}^{m}e^{2\pi is_{j}k}\right|\right).\end{aligned}}\qquad (1)

Cette inégalité vaut pour des entiers naturels arbitraires m,n, et donne une forme quantitative du critère de Weyl pour l'équidistribution.

Une variante multidimensionnelle de (1) est connue sous le nom d'inégalité d'Erdős–Turán–Koksma.

Références

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Erdős–Turán inequality » (voir la liste des auteurs).

↑ Erdős et Turán, « On a problem in the theory of uniform distribution. I. », Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, vol. 51,‎ 1948, p. 1146–1154 (MR 0027895, zbMATH 0031.25402, lire en ligne)
↑ Erdős et Turán, « On a problem in the theory of uniform distribution. II. », Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, vol. 51,‎ 1948, p. 1262–1269 (MR 0027895, zbMATH 0032.01601, lire en ligne)

Bibliographie

Glyn Harman, Metric Number Theory, vol. 18, Clarendon Press, coll. « London Mathematical Society Monographs. New Series », 1998 (ISBN 0-19-850083-1, zbMATH 1081.11057)

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