On the lattice discrepancy of bodies of rotation with boundary points of curvature zero

Abstract

This article gives an asymptotic result for the lattice point discrepancy of a large body of rotation in R 3 , whose boundary is piecewise smooth and contains points of vanishing Gaussian curvature.

The lattice discrepancy of certain three-dimensional bodies

“…We balance this against the first term on the right hand side of (6.12) -except for the powers of 2 r , with respect to which we simply compensate the factor 2 r mk−27 28mk in the next-to-last term of (6.12). This gives H (V r ,j) = 2 −j 18k−11 25 Obviously H (V r ,j) > 1 throughout: As before after (6.10),…”

“…For partial results, see Haberland [4], Krätzel [14], [16], [17], [18] and Peter [26]. A fairly general theorem on bodies of rotation has been established by Nowak [25]. The method used there can be called the "cut-into-slices approach": Since xB intersects every plane orthogonal to its axis of rotation in a circular disc, a very accurate approximation to the latter's discrepancy is employed, followed by a ("careful") summation with respect to the third coordinate.…”

Lattice points in bodies of rotation with a dent

“…In the sequel, Krätzel [16][17][18] established a series of results, both on quite special and on fairly general three-dimensional bodies. Furthermore, the author [24] obtained a rather general theorem for bodies of rotation d with boundary points of curvature zero.…”

The lattice point discrepancy of a torus in ℝ3

“…For the case that ∂R is of genus zero, this situation has been worked out in articles by the author [17], [18]. A recent joint paper with E. Krätzel [15] deals with the convex body R k : (x 2 + y 2 ) k/2 + |z| k 1, k > 2 xed, which is generated by the rotation of a Lamé's curve about one coordinate axis.…”