Repairing the refined-decoupling proof of the 5/4 planar pinned Falconer theorem

Abstract

We show that the literal arbitrary-packet form of the refined-decoupling estimate printed in the Guth--Iosevich-Ou-Wang proof of the planar pinned Falconer theorem is false, even after its tube dimensions are normalized. An explicit collar construction gives a fixed-power counterexample: all packets are active on one square while none of their smaller labelled tubes meets that square. We then give a non circular repair of the original proof route. The analytic input is an enlargement-stable arbitrary-packet theorem in which a packet is concentrated on an $a$-dilate and multiplicity is counted with a strictly larger $b$-dilate. We prove this theorem directly from weighted $\ell^2$ decoupling by an induction that tracks the dilation margin through parabolic rescaling. The corrected theorem applies directly to the original Falconer parent packets after an exact frequency truncation and an absolute small-packet cutoff; no parent-to-canonical decomposition is needed. We then rebuild the good-tube incidence estimate, retain the neighborhood forced by local constancy, and supply a uniform regularization and limiting argument. The principal frequency exponent remains $-(\alpha+1)/3$, so the energy argument closes exactly for $\alpha>5/4$. The pinned theorem itself is not contradicted and is also known through later microlocal methods. A later canonical wave-packet treatment of refined decoupling overlaps with the activity-tube viewpoint but not with the counterexample or the repaired Falconer proof chain.