By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is $\Lambda \left(p\right)$ for all $p$ but is not a Rosenthal set. This holds also for the sequence of primes.