Multiplicative Inequalities for the $L_1$ Norm: Applications in Analysis and Number Theory

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 $[n^\alpha]$ with $1\le\alpha <\infty$, are developed. A new combinatorial theorem is proved that makes it possible to estimate the arithmetic characteristics of spectra under fairly general assumptions.