Volumes, areas and other Falconer-type problems - Silva, Belmiro Galo da (2021)
In this thesis, we investigate the Falconer-type problems about point configurations and in different dimensions. It is well-known the concept of the Hausdorff measure is a generalization of the Lebesgue measure and the Falconer distance problem aims to relate these two topics when it asks how large does the Hausdorff dimension of a compact set need to be to ensure the Lebesgue measure of the distance set. In the first moment, we consider a k-point configurations in Rd and we prove that a compact set E EUR Rd determines a positive measure of such volume types if the Hausdorff dimension of E is greater than d d1 2kd generalizing some results in this field. This portion of the work represents joint work with Dr. Alex McDonald. In the second moment, we study a Falconer-type problem on a 4-point configuration in the plane and we prove that a compact set E EUR R2 determines a positive measure of such Galo area types if the Hausdorff dimension of E is greater than 3 2 extending some results from A. McDonald in [22].