The lattice discrepancy of bodies bounded by a rotating Lamé’s curv

The lattice point counting problem on the Heisenberg groups

“…[27], [28], [18], [19], [21], [22]. As we shall show in §6, our method, when applied to the Euclidean lattice point counting problem in some of these bodies, actually yields the same (main) error estimate obtained for some of them in the references cited.…”

“…These include the unit balls of p -norms on R n and some generalizations, and the effect of vanishing curvature on the error estimates have been investigated extensively in e.g. [18], [19], [20], [21], [22], [23], [25], [26], [27], [28], [29] and [30].…”

Higher order derivative tests for exponential sums incorporating the Discrete Hardy–Littlewood method

“…A special feature of the present problem is that Theorem 2 will be applied with r = 8, thus derivatives of fairly high orders have to be computed and bounded away from zero. Similarly, one can improve the bounds for the lattice discrepancy of other bodies of rotation, like of that bounded by a rotating Lamé's curve [10] and of the torus in R 3 [12].…”

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.…”

“…This appears quite natural and, at the same time, furnishes a much more precise result. The technique used is inspired by the papers [1] and [15]. However, ultimately the skillful handling of multiple exponential sums can be traced back to I. M. Vinigradov [21] who in this way obtained the bound O t 11/8+ε for the lattice discrepancy of the three-dimensional sphere.…”