Point configurations inside thin subsets of Euclidean space
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.