The Geometry of Diophantine Inequalities

22 January 2014
Simon Rydin-Myerson
A major project in number theory runs as follows. Suppose some Diophantine equation has infinitely many integer solutions. One can then ask how common solutions are: roughly how many solutions are there in integers $\in [ -B, \, B ] $? And ideally one wants an answer in terms of the geometry of the original equation. What if we ask the same question about Diophantine inequalities, instead of equations? This is surely a less deep question, but has the advantage that all the geometry we need is over $\mathbb{R}$. This makes the best-understood examples much easier to state and understand.