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Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory

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Abstract

The paper is devoted to multiplicative lower estimates for the L 1 norm and their applications in analysis and number theory. Multiplicative inequalities of the following three types are considered: martingale (for the Haar system), complex trigonometric (for exponential sums), and real trigonometric. A new method for obtaining sharp bounds for the integral norm of trigonometric and power series is proposed; this method uses the number-theoretic and combinatorial characteristics of the spectrum. Applications of the method (both in H 1 and L 1) to an important class of power density spectra, including [n α] with 1 ≤ α < ∞, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.

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  1. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    S. V. Bochkarev

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  1. S. V. Bochkarev
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Original Russian Text © S.V. Bochkarev, 2006, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 55–70.

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Bochkarev, S.V. Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory. Proc. Steklov Inst. Math. 255, 49–64 (2006). https://doi.org/10.1134/S0081543806040055

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