Pinned dot product set estimates
Abstract
We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets A?Rn and a,x?Rn, we study sets of the form (Formula presented.) We give a picture of the current state of the art by discussing what is known, and we prove some new results and special cases. We obtain lower bounds on the Hausdorff dimension of A to guarantee that ?xa(A) is large in some quantitative sense for some a?A (i.e.,?xa(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.
Department(s)
Mathematics
Document Type
Article
DOI
10.1007/s40687-025-00583-x
Publication Date
3-1-2026
Recommended Citation
Senger, Steven; Bright, Paige; and Marshall, Caleb, "Pinned dot product set estimates" (2026). Faculty Scholarship. 27.
https://bearworks.missouristate.edu/articles00/27
Journal Title
Research in Mathematical Sciences
