Title:
Point configurations inside thin subsets of Euclidean space
Abstract:
We are going to explore the following basic question in geometric measure theory.
How large does the Hausdorff dimension of a subset of Euclidean space need to be to
ensure that it contains: i) an equilateral simplex ii) a positive proportion of all
equilateral simplexes of a given dimension iii) more general geometric configurations.
All the proofs involve an interplay of Fourier analytic and combinatorial methods.
Connections with geometric combinatorics will also be discussed.
