  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 $L^p_E(T)$
and $C_E(T)$ of functions on the circle and express block unconditionality as
arithmetical conditions on $E$. Our work shows that the spaces $L^p_E(T)$, $p$
an even integer, have a singular behaviour from the almost isometric point of
view: property UMAP does not interpolate between spaces $L^p_E(T)$ and
$L^{p+2}_E(T)$. These arithmetical conditions are used to construct
counterexamples for several natural questions and to investigate the maximal
density of such sets $E$. We also prove that if $E=\{n_k\}_{k\ge1}$ with
$|n_{k+1}/n_k|\to\infty$, then $C_E(T)$ has UMAP and we get a sharp estimate of
the Sidon constant of Hadamard sets. Finally, we investigate the relationship
of metric unconditionality and probability theory.
