A relatively recent problem which has grown in interest in the past few decades is, given a subset of Euclidean space, what conditions does it need to have for certain geometric conditions to apply? Results from Falconer (1985) and later Erdogan (2005) gave estimates for how large a subset needs to be for the distance set to have positive lebesgue density. Other results such as Ziegler (2006) and Chan, Laba, Pramanik (2016) gave estimates for how large a subset needs to contain certain geometric configurations.

This talk will discuss an improvement on the result by Iosevich and Liu (2016) which recovered equilateral triangles in subsets of \(R^d\) with sufficiently large hausdorff dimension. We improve upon this result by adapting the mechanism to apply to isosceles triangles of angle \(\theta \in (0,\pi)\). While this result is still less general than that of Iosevich and Magyar (2020), there is potential hope for further generalization and computation of the currently unknown dimensional constant.