A mean-square bound for the lattice discrepancy of bodies of rotation with flat points on the boundary

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

“…The case that the flat points on the axis of rotation are the only boundary points of curvature zero has been dealt with by the author in a previous article [21]. In that paper the main issue was to show that, in this case, ∆ R (t) is small "on average", i.e., that (under slightly more stringent smoothness conditions on ∂R)…”

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.…”

A mean-square bound concerning the lattice discrepancy of a torus in ℝ3

“…A paper by the second named author [11] deals with bodies of rotation B where the curvature vanishes at the two points of intersection of ∂B with the axis of rotation. It turns out that these two points contribute to the lattice discrepancy an amount of exact order O(t 2−2/(N +2) ), if the curvature of the meridian curve (whose rotation generates ∂B) is assumed to vanish of order N .…”