Lattice points in large convex bodies

Denote by A(Bs) (x) the number of points of the lattice Z(s) in the "blown up" domain xB(s), where B-s is a convex body in R-s (s greater than or equal to 3) whose boundary is smooth and has nonzero curvature throughout. It is proved that for every fixed epsilon > 0 A(Bs) (x) = vol(B-s)x(s) + O(x(s-2+lambda(s)+epsilon)), where lambda(s) = (s + 4)/(s(2) + s + 2) for s greater than or equal to 5, lambda(4) = 6/17 and lambda(3) = 20/43. This improves a classic result of E. HLAWKA [8] and its refinements due to E. KRATZEL and W. G. NOWAK ([14], [15]). The proof uses a multidimensional variant of the method of van der Corput for the estimation of exponential sums.