Related papers: On two lattice points problems about the parabola

We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits…

Suppose $a,~b$ are fixed algebraic numbers with $1\leq a<b$. Let $\Delta_{a,b}(x)$ be the error term for the number of lattice points in a two-dimensional area $h^ar^b\leq x $ with $h, r$ positive integers. In this paper, we establish an…

We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in $\Omega\times (0,T)$ -…

We obtain new omega results for the error terms in two classical lattice point problems. These results are likely to be the best possible.

We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves…

Let $\Gamma$ be an irreducible lattice in $\PSL_2(\RR)^d$ ($d\in\NN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances ${\bf D}(\Gm,z)\subset \RR^d$ defined in (1).…

We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums…

The iterated composition of two operators, both of which are involutions and translation invariant, partitions the set of lattice points in the plane into an infinite sequence of discrete parabolas. Each such parabola contains an associated…

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…

For d-dimensional irrational ellipsoids E with d >= 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(r^{d-2}). The estimate refines an earlier authors'…

We present new estimates for sums of the divisor function, and other similar arithmetic functions, in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an…

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses…

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be…

We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Kor$\acute{a}$nyi Heisenberg-norm…

The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity…

We study an optimal stretching problem, which is a variant lattice point problem, for convex domains in $\mathbb{R}^d$ ($d\geq 2$) with smooth boundary of finite type that are symmetric with respect to each coordinate hyperplane/axis. We…

We prove a sharp upper bound on the number of boundary lattice points of a rational polygon in terms of its denominator and the number of interior lattice points, generalizing Scott's inequality. We then give sharp lower and upper bounds on…