Distributive Aronszajn trees

Ben-David and Shelah proved that if λ is a singular strong-limit cardinal and 2 ^λ = λ ⁺ , then ^∗ _λ entails the existence of a normal λ-distributive λ ⁺ -Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis ^∗ _λ by (λ ⁺ ,<λ). As (λ ⁺ ,<λ) does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for κ regular uncountable, (κ) entails the existence of a partition of κ into κ many fat sets. When contrasted with a classical model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that ω2 cannot be split into two fat sets.