On approximate ℓ₁ systems in Banach spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i = ₁ in X which satisfy ∥∑_{i∈ A} ± x_i∥ ≥ A - f(A), for all finite A⊂ ∌ and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.