Lattice points in rational ellipsoids

Abstract

We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.

Lattice points in bodies of rotation with a dent

“…We add the remark that for the special case of the unit ball B 0 in R 3 ("sphere problem"), the much better estimate D(B 0 ; t) = O(t 21/16+ε ) is known, due to Heath-Brown [7]. Furthermore, the case of ellipsoids has been considered by Bentkus and Götze [1], Chamizo, Cristobal and Ubis [3], Krätzel and the author [21], and the author [25,26].…”

The sphere problem and the L-functions

“…It is not difficult to prove that θ 3 1, in fact it is known [24] that S(R) − 4π 3 R 3 is not o R(log R) 1/2 (the same holds for visible points [2], and for general convex bodies with a smaller logarithmic power [19]). In general, upper bounds are conjecturally less precise.…”

“…Shortly after, Heath-Brown [11] improved this bound to θ 3 21/16 using his important work [10] on large sieve inequalities for real characters. In a recent work [3] the authors and Ubis have introduced a new complementary term involving Fourier coefficients of automorphic forms that allows to treat the case of rational ellipsoids.…”

On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

“…Most of them are closely related to the determination of lattice points in circles [43], ellipsoids [22,45], or surfaces of revolution [19].…”