Breadcrumb
Counting lattice points and o-minimal structures
Wed 30 January 2013, 16:00
Martin Widmer
Royal Holloway
Organisers: Tim Dokchitser, Daniel Loughran
ABSTRACT
Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and let $Z\subseteq \mathbb{R}^{m+n}$ be a parameterised family of subsets $Z_T$ of $\mathbb{R}^n$. We are interested in the cardinality $|\Lambda\cap Z_T|$. Using o-minimal structures from model theory we prove for fairly general families $Z$ an estimate, which is essentially best possible. To this end we show that the volumes of the projections of $Z_T$ to an arbitrary $j$-dimensional subspace are bounded in terms of the volumes of the projections to the $j$-dimensional coordinate spaces and the family $Z$, but independently of $T$.
This is joint work with Fabrizio Barroero.