On volumes determined by subsets of Euclidean space
Allan Greenleaf, Alex Iosevich, Mihalis Mourgoglou ยท Forum Mathematicum ยท 2013
Abstract Given E โ โ d , define the volume set of E, ๐ฑ(E) = {det(x 1,x 2,...,xd ) : xj โ E}. In โ3, we prove that ๐ฑ(E) has positive Lebesgue measure if either the Hausdorff dimension of E โ โ3 is greater than 13/5, or E is a product set of the form E = B 1รB 2รB 3 with Bj โ โ, dimโ(Bj ) > 2/3, j = 1,2,3. We show that the same conclusion holds for ๐ฑ(E) of Salem subsets E โ โ d with dimโ(E) > d - 1, and give applications to discrete combinatorial geometry.