Radial Projections in Rⁿ Revisited

We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given X,Y⊂ Rⁿ Borel sets and X≠ ∅. If Y ∈ (k,k+1] for some k∈ \1,, n-1\, then \[ x∈ X π_x(Y \{x\}) ≥ min \{ X + Y - k, k\}. \] Our results give a new approach to solving a conjecture of Lund-Pham-Thu in all dimensions and for all ranges of Y. The first of our two methods for proving the above theorem is shorter, utilizing a result of the first author and Gan. Our second method, though longer, follows the original methodology of Orponen--Shmerkin--Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg-set estimate for lines. These new estimates may be of independent interest.