Pinned Dot Product Set Estimates

We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset {\mathbb{R}}^n$ and $a,x\in {\mathbb{R}}^n$, we study sets of the form $${\mathrm{\Pi }}_x^a(A):=\{\alpha \in \mathbb{R}:(a-x)\cdot y=\alpha ,\text{ for some }y\in A\}\mathrm{.}$$ We discuss some of what is already known to give a picture of the current state of the art, as well as prove some new results and special cases. We obtain lower bounds on the Hausdorff dimension of $A$ to guarantee that ${\mathrm{\Pi }}_x^a(A)$ is large in some quantitative sense for some $a\in A$ (i.e. ${\mathrm{\Pi }}_x^a(A)$ has large Hausdorff dimension, positive measure, or nonempty interior). Our approach to all three senses of “size” is the same, and we make use of both classical and recent results on projection theory.