Orthogonal projection, dual Furstenberg problem, and discretized sum-product

In this paper we come up with a dual version of the Furstenberg problem and obtain partial results via Lᵖ estimates of orthogonal projections. Examples are also discussed. Moreover, compared with general sets, we find that special structure like Cartesian product has better Lᵖ-behavior. This leads to improvement on some discretized sum-product estimates.