Homogeneous Form Inequalities (II): Lattice Point Counting

This is the second part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, and among other results, sharp estimates for the volume of the semialgebraic domain under consideration were established in the first part. Here, these estimations are employed to establish (a set of) statements providing a global count on the number of solutions to the Diophantine inequalities under consideration. The results thus obtained go beyond the usual assumptions of smoothness and of nonvanishing of the Gaussian curvature. Specifically, the introduction of a semialgebraic level of flatness emerging from geometric tomography enables one to characterise the cases when an accurate counting estimate can be obtained in thin neighbourhoods of an algebraic variety. This is done with a specific view towards applications to Diophantine approximation.