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We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_E^p\left(\right)$ and $C_E\left(\right)$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_E\left(\right)$, p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_E^p\left(\right)$ and $L_E^{p+2}\left(\right)$. These...

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.