Abstract

In 1929 Oppenheim asked whether the image $Q(\mathbb{Z})$ of an indefinite, non-degenerate quadratic form $Q$ in more than $5$ variables, which is not proportional to a rational quadratic form is dense in $\mathbb{R}$. This question came to be known as the Oppenheim conjecture and was answered in 1986 by Margulis, who used methods coming from homogeneous dynamics to show that in fact $3$ variables suffice. Unfortunately, these methods don’t readily apply to hypersurfaces of higher degrees.

In 1946, Davenport and Heilbronn used the duality between $\mathbb{R}$ and itself and introduced a new variant of the circle method, which allowed them to prove that the integer image of a special polynomial is dense in $\mathbb{R}$. We present a refinement of their method by Freeman and show how it can be applied to solve counting problems connected to denseness results and Diophantine inequalities. This talk is intended for an audience with no prior experience with circle method and will include a gentle introduction to classical theory.