A common motif in my research is the dichotomy between structured and random situations across combinatorics, probability, and analytic number theory. This proposal highlights two instantiations of this dichotomy. The first concerns the local limit distributions of low–degree polynomials and identifying structured for the failure of local limit theorems. This perspective has been used in a host of my works and I aim further these techniques to explore universality at microscopic scales. The second concerns the structure of integer sets which can be categorized as either “random–like” or “algebraically structured”. My work has improved the quantitative aspects of this connection leading to improved results in additive combinatorics (in particular Szemerédi’s theorem) and analytic number theory. This proposal seeks to extend these advances to multidimensional and polynomial patterns and to apply these tools to tackle problems in analytic number theory.