Geometric measure theory attempts to identify finite point configurations in sufficiently large subsets of the Euclidean space. If these subsets are assumed to have a positive Lebesgue measure, then the corresponding results are sometimes called Euclidean density theorems and their study was initiated in the 1980s. Results of this type are particularly prone to the tools from the harmonic analysis, such as multilinear singular and oscillatory integrals.
I am grateful to the SCS 2022 organizers and participants for numerous useful remarks on the presentation.
A few recent references mentioned on the slides:
Durcik and K. (2018): Boxes, extended boxes, and sets of positive upper density in the Euclidean space, Math. Proc. Cambridge Philos. Soc. 171 (2021), no. 3, 481-501.
Durcik and K. (2020): A Szemerédi-type theorem for subsets of the unit cube, Anal. PDE. 15 (2022), no. 2, 507-549.
K. (2020): Density theorems for anisotropic point configurations, accepted for publication in Canad. J. Math.
Falconer, K., and Yavicoli (2020): The density of sets containing large similar copies of finite sets, accepted for publication in J. Anal. Math.
Durcik and Stipčić (2022): Quantitative bounds for product of simplices in subsets of the unit cube, preprint. https://arxiv.org/abs/2206.10004