Integer Points in Large Bodies

Variance of lattice point counting in some special shells in $\mathbb{R}^d$

“…Many authors made efforts to study general domains under different curvature assumptions of the boundary ∂B. We refer interested readers to Ivić, Krätzel, Kühleitner and Nowak [10] and Nowak [15] which gave excellent overview of the development of this problem.…”

Lattice points in stretched model domains of finite type in Rd

“…Even for the unit disk in the plane, in which case the problem is the famous Gauss circle problem, we are still far away from the conjectured optimal growth rate. For historical results on the lattice point problem, see for example [10], [17], etc. Motivated by the "eigenvalue optimization among rectangles" problem in spectral geometry, in the paper [1] P. Antunes and P. Freitas studied the following variation of the Gauss circle problem: find the "optimal stretching factor" that maximizes the remainder term #(N 2 ∩ tAB) − πt 2 /4, ( * )where B = B(0, 1) is the unit disk in R 2 , and A = diag(s, s −1 ).…”

“…In fact, even the classical lattice point problem for domains having boundary points of vanishing Gaussian curvature is not well understood, especially in high dimensions. For partial results we refer interested readers to [10], [17], [8], [9], and the references given there.…”

Approximation and convergence of the intrinsic volume

“…Grouping the k-planes in 1-parameter families, we are interested in the case = k + 1, for which we get s −k = s 1 = 2 in the denominator. Setting ϕ(E) = χ(M ∩ E), we first use (18) and then (9) to rewrite (8) as…”

“…Letting #(M ∩ tZ 3 ) be the number of points of the dilated integer grid in M, it is well known that t 3 #(M ∩ tZ 3 ) converges to Vol(M) as t goes to zero. The central question of the classic lattice point theory, as founded by E. Landau and others in the first decades of the 20th century, is to estimate the lattice discrepancy, which is defined as t 3 #(M ∩ tZ 3 ) − Vol(M); see the recent survey [18] for more details. Since the lattice discrepancy vanishes as t goes to zero, we may approximate M with #(M ∩ tZ 3 ) cubes of edge length t whose centers are in M ∩ tZ 3 , such that the volume is preserved asymptotically, as t goes to zero.…”