Commensurate sequences of characters

Abstract:

If $({a_j})$ and $({b_j})$ are sequences of characters on compact abelian groups $S$ and $T$ respectively such that for every sequence of scalars $({\alpha _j})||\sum {\alpha _j}{a_j}|{|_\infty } \asymp ||\sum {\alpha _j}{b_j}|{|_\infty }$ tnen for every $1 \leq p < \infty$ and every sequence $({x_j})$ of elements of an arbitrary Banach space $X$ \[ {\int _S {\left \| {\sum {{x_j}{a_j}} } \right \|} ^p}ds \asymp {\int _T {\left \| {\sum {{x_j}b} } \right \|} ^p}dt.\] This result generalizes a result of Pisier [Pi 1] for Sidon sets. For topological Sidon sets on ${\mathbf {R}}$ a slightly stronger result holds.