Sharpness of the 3/4 exponent in Wolff’s incidence bound for curvilinear rectangles

Establish whether the exponent 3/4 in the incidence bound for pairwise incomparable (η,t)-rectangles that are (μ,ν)-rich with respect to a t-bipartite pair of curves—namely, the inequality #ℛ ≲_ε ( #𝒞 #𝒟 )^ε [ ( #𝒞 #𝒟/(μν) )^{3/4} + #𝒞/μ + #𝒟/ν ] (originally Wolff’s Lemma 1.4, and stated here as an incidence bound for curvilinear rectangles)—is optimal, or whether it can be improved to an exponent strictly less than 3/4.

Background

The proof of the planar bilinear Kakeya estimate in this paper relies on a rectangle-counting lemma due to Wolff (and its formulation for curvilinear rectangles), where the key exponent 3/4 controls the dominant term. The authors note that Wolff believed 3/4 to be the best achievable with his techniques and that no explicit improvement is known.

Improving this exponent would have immediate implications for the bilinear Kakeya estimates considered here and could potentially lower the L^p exponent in the Heisenberg setting.