Falconer estimate for circular and cinematic Furstenberg problem

Abstract

We prove the Falconer-type bound $\dim_{\mathrm H}F\geq u+1$ for planar circular $(u,v)$-Furstenberg sets whenever $0<u\leq1$, $0<v\leq3$, and $2u+v\geq3$. The same bound holds for smooth cinematic curve families. This resolves the circular Furstenberg conjecture of Green, Harris, Ou, Ren, and Tammen in its full Falconer range $2u+v\geq3$.