Vishnu Priya Singh | Searcharxiv

arXiv2026

Inverse Theorems for Point-Sphere Incidences over Finite Fields

Let Q be a nondegenerate quadratic form on $\mathbb F_q^d$, with q odd & $d\ge2$. We prove a two-scale inverse theory for point-sphere incidences. At the maximal-incidence endpoint, put $μ=1-I(P,S)/(|P||S|)$. If $μ<(d+1)/(d+2)^2$, then deleting at most a proportion $\sqrt{(d+1)μ}$ from each side places the retained configuration in one explicit lifted-linear model: the sphere parameters $Φ(S_Q(c,r))=(2c,r-Q(c))$ lie in an affine space L of dimension at most d, & the retained points lie in its common affine-quadric base. The natural deletion distance satisfies $μ\le dist_{lin}(P,S) \le2\sqrt{(d+1)μ}$ and the square-root exponent is optimal. Under codimension-two nonconcentration, & provided at least two spheres survive, the model reduces to one coaxal pencil. This classification has a sharp lower-scale boundary. A polynomial-root chart on one lifted rational normal curve realizes every admissible d-uniform hypergraph; consequently any list of individual point-set templates approximating all resulting constant-K systems within $o(q^d)$ edits has size $\exp(Ω_d(q^d))$. Exact fixed-nonzero-radius centered direct sums occur even when $|S|\ge4q/K^2$, & a prescribed-radius twisted-cubic system in dimension three has a complete rich-pair graph but neither a heavy pair section nor a quantitatively nontrivial centered block decomposition. At the fixed-radius deviation scale, what remains universally true is sharp adaptive extraction. If $σ(P,S)\ge Kq^{(d-1)/2}\sqrt{|P||S|}$ & $K^2q^{d-1}|S|\ge4|P|$, then at least $K^2|S|/4$ ordered pairs have a common P-section of size at least $K^2q^{d-1}/(4|S|)$; for $d\ge3$ the witness count improves to $K^2q|S|/8$. We prove mixed-radius profile stability for both signs, a log-free refinement, a cap-free positive-surplus decomposition, & applications to affine hyperplanes, pinned distances, dot products.

math.CO↗

arXiv2026

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

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 $-(α+1)/3$, so the energy argument closes exactly for $α>5/4$. 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.

math.CA↗