Counting lattice points and o-minimal structures

Let Λ be a lattice in Graphic, and let Graphic be a definable family in an O-minimal structure over Graphic. We give sharp estimates for the number of lattice points in the fibers Graphic. Along the way, we show that for any subspace Graphic of dimension j>0 the j-volume of the orthogonal projection of ZT to Σ 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.