Submission date: 22 October 2012.

Abstract:

Let \Lambda be a lattice in R^n, and let Z \subseteq R^{m+n} be a definable family in an o-minimal structure over R. We give sharp estimates for the number of lattice points in the fibers Z_T={x in R^n: (T,x) in Z}. Along the way we show that for any subspace \Sigma \subseteq R^n of dimension j>0 the j-volume of the orthogonal projection of Z_T to \Sigma is, up to a constant depending only on the family Z, bounded by the maximal j-dimensional volume of the orthogonal projections to the j-dimensional coordinate subspaces.

Mathematics Subject Classification: 11H06, 03C98, 03C64 (Primary) 11P21, 28A75, 52C07 (Secondary)

Keywords and phrases:

Full text arXiv 1210.5943: pdf, ps.