  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 $L^p$ norm ($p\geq 1$) and when $r$ is large enough,
the $L^p$ norm of such a linear combination is equivalent to the $\ell^p$ norm
of the sequence of coefficients. In other words, one can describe many ways of
embedding $\ell^p$ into $L^p$ based on Fourier coefficients. This generalizes
to vector valued $L^p$ spaces.
