The lattice discrepancy of certain three-dimensional bodies

Abstract

Based on a very precise approximation to the lattice discrepancy of a Lamé disc, an asymptotic formula is established for the number of lattice points in the three-dimensional bodyfor large real x and fixed reals m, k . Particular attention is paid to the boundary points of Gaussian curvature zero. (2000): 11P21, 11N37, 11K38, 52C07. Mathematics Subject Classification

Exponential and Infinitary Divisors

“…, a k ; n). (See [11] for the form of the main term.) For the sake of brevity denote ∆ k (x) = ∆(1, .…”

“…Ω-estimate of the error term E(x) follows again from [14]. To obtain E(x) ≪ ≪ x 8/19 we use [11,Th. 6.8], which implies θ(1, 2, 2, 2, 2, 2, 2, 2, 2)…”

A note on lattice points in model domains of finite type in $${\mathbb{R}^d}$$ R d

“…For further results of super spheres (ellipsoids) see [6,4] and the references contained therein. Krätzel [7] and Krätzel and Nowak [8,9] study a special class of convex domains in R 3 ,…”

“…with certain assumptions on reals k and m (for example, in [9], k > 2, m > 1, and mk ≥ 7/3). The contribution of flat points is evaluated precisely and that of other boundary points is estimated.…”

“…If we take that d = 3, n = 2, m 1 = m 2 = m, ω 1 = ω 2 = ω 3 = k, and d 1 = 1, the domain D is in the special form of (1.3). Since we only consider domains with smooth boundary we did not allow exponents m and k to be real numbers, to the contrary Krätzel and Nowak [8,9] do allow such general exponents. We mainly use harmonic analysis tools, while Krätzel and Nowak apply a "cut-into-slices"-method to reduce a three-dimensional problem to a two-dimensional one and then work carefully on the latter problem.…”