In 1986, Falconer conjectured that for any compact set E ⊂ Rd (d ≥ 2) whose Hausdorff dimension is greater than d/2 , its distance set

∆(E) = {d(x, y) : x, y ∈ E}

must have positive Lebesgue measure. The best currently known dimensional threshold is d/2 + 1/3 , due to Wolff on the plane and Erdogan in higher dimensions. In this talk we will discuss about different approaches to this problem and its generalizations. Another problem I will talk about is the existence of a given pattern contained in E ⊂ Rd. More precisely, we prove that when d ≥ 4, any set E ⊂ Rd of large Hausdorff dimension must contain vertices of an equilateral triangle.