Lattice points in large convex bodies

Abstract

In this article we investigate the number A (t) of lattice points in x/t ~ where is a convex body in R s (s/> 3) which has a smooth boundary with nonzero Gaussian curvature throughout, and t is a large real parameter. We establish an asymptotic formula A (t) = Vt s/2 + 0 (t x(s~) (V the volume of ~) which improves upon a classic result of E. HLAWXA [5].

Mean square discrepancy bounds for the number of lattice points in large convex bodies

“…For example, Jarník [14] established the bound O(t −d/2+ε ) for the relative error, with any ε > 0, for almost all ellipsoids with axes parallel to the coordinate axes. For general convex domains with non-vanishing curvature on the boundary, W. Müller [22] proved that ∆ Ω (t) = O(t −2+λ(d)+ε ), where λ(d) = (d + 4)/(d 2 + d + 2), if d ≥ 5, λ(4) = 6/17 and λ(3) = 20/43, improving on earlier results by Krätzel and Nowak [19]. For planar domains, Huxley [11] obtained this estimate with λ(2) = 46/73, which implies the relative error O(t −100/73 (log t) 315/146 ).…”

Lattice Points in Three-Dimensional Convex Bodies

“…There are some results for the number of lattice points in convex bodies provided that for all points on the bound the Gaussian curvature is positive throughout. See the papers of E. Hlawka [4] and W. G. Nowak and the author [8], [9]. K. Haberland [3] considered also the case that on the bound are isolated points with Gaussian curvature zero provided that the tangential plane is rational.…”

Two problems associated with convex finite type domains

“…. This estimate however is not sharp, and several authors beginning with van der Corput have obtained improvements for the case of nonvanishing Gauß curvature; see the monographs by Krätzel [18] and Huxley [14], and in particular the papers by Krätzel and Nowak [20] and recent improvements by W. Müller [22] for results on general convex bodies with nonvanishing curvature in higher dimensions. In [24, I], [25] Randol obtained better estimates for the case of convex domains in the plane with finite type boundary; these are sharp for Ω = {x :…”