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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