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Jun 2, 2013 at 15:50 comment added Sungjin Kim I think the true bound is $S(r)=O(r^2\log r + r^3)$ now.
Jun 2, 2013 at 15:49 comment added Sungjin Kim I realized that that estimate was overestimate. This is a counting argument for lattice points, like one uses for Gauss Circle Problem. The number of lattice points inside a convex region is approximated by the area of the region + error term depending on perimeter.
Jun 2, 2013 at 10:18 comment added Anton Menshov Could you clarify how did you get that $A_t \leq 4\sqrt{2} r^3 t + O(r^3)$ ?
Jun 1, 2013 at 23:19 comment added Sungjin Kim Argument seems to work in general, $S(r)=O(r^{n^2-1}\log r)$.
Jun 1, 2013 at 23:01 history answered Sungjin Kim CC BY-SA 3.0