Continuum Beck-Type Problems

Radial projections have been a quickly developing topic in harmonic analysis. The work of Orponen-Shmerkin-Wang connected this topic to a continuum Beck theorem for lines, finding a (lower) bound for the dimension of lines that contain at least two points in a given subset of $\mathbb{R}^n$. Since then, results of Ren and B.-Marshall, have studied similar Beck-type theorems in higher dimensions. The work thus far has made use of (dual) Furstenberg sets. We discuss these results as a way to motivate dual Furstenberg-type problems, including a new problem of this type that has come out of ongoing work of B.-Ortiz-Zakharov.