Bounds on moments of weighted sums of finite Riesz products

Abstract: Let $n_j$ be a lacunary sequence of integers, such that $n_{j+1}/n_j\geq r$. We are interested in linear combinations of the sequence of finite Riesz products $\prod_{j=1}^N(1+\cos(n_j t))$. We prove that, whenever the Riesz products are normalized in $L<sup>p$ norm ($p\geq 1$) and when $r$ is large enough, the $L<sup>p$ norm of such a linear combination is equivalent to the $\ell<sup>p$ norm of the sequence of coefficients. In other words, one can describe many ways of embedding $\ell<sup>p$ into $L<sup>p$ based on Fourier coefficients. This generalizes to vector valued $L<sup>p$ spaces.