Pinned dot product sets are large when the underlying set has Hausdorff dimension above n/2 with spread translations, or above (n+1)/2 for any translation, proven by reducing dot products to orthogonal projections.
Discretized Radial Projections in $\mathbb{R}^d$
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abstract
We generalize a Furstenberg-type result of Orponen-Shmerkin to higher dimensions, leading to an $\epsilon$-improvement in Kaufman's projection theorem for hyperplanes and an unconditional discretized radial projection theorem in the spirit of Orponen-Shmerkin-Wang. Our proof relies on a new incidence estimate for $\delta$-tubes and a quasi-product set of $\delta$-balls in $\mathbb{R}^d$.
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Pinned Dot Product Set Estimates
Pinned dot product sets are large when the underlying set has Hausdorff dimension above n/2 with spread translations, or above (n+1)/2 for any translation, proven by reducing dot products to orthogonal projections.