Integer points on the dilation of a subanalytic surface

Let Ω ⊂Rn be a compact subanalytic set of dimension 2 and t≥1. This paper gives an upper bound as t→∞ for the number of integer points on the homothetic dilation tΩ of Ω that do not reside on any connected semialgebraic subset of tΩ of positive dimension. Implications for the density of rational points on Ω are also elaborated.