Lattice Point Problem Research Articles

Lattice point problems and values of quadratic forms

For d-dimensional ellipsoids E with d≥5 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 \(\mathcal{O}(r^{d-2})\) for general ellipsoids and up to an error of order o(rd-2) for irrational ones. The estimate refines earlier bounds of the same order for dimensions d≥9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],m∈ℤ d of positive definite irrational quadratic forms Q of dimension d≥5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.

On a lattice point problem arising in the spectral analysis of periodic operators

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

Short rational generating functions for lattice point problems

We prove that for any fixed $d$ the generating function of the projection of the set of integer points in a rational $d$-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers $a_{1}, \ldots , a_{d}$) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.

Omega results for the divisor and circle problems

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

A third derivative test for mean values of exponential sums with application to lattice point problems

A third derivative test for mean values of exponential sums with application to lattice point problems

On the recovery of a function on a circular domain

We consider the problem of estimating a function f (x, y) on the unit disk f {(x, y): x/sup 2/+y/sup 2//spl les/1}, given a discrete and noisy data recorded on a regular square grid. An estimate of f (x, y) based on a class of orthogonal and complete functions over the unit disk is proposed. This class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate. For-radial functions, the orthogonal set has a particularly simple form being related to the classical Legendre polynomials. We give the statistical accuracy analysis of the proposed estimate of f (x, y) in the sense of the L/sub 2/ metric. It is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain. This is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle. In fact, the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain.

A path integral approach to lattice point problems

The main problem in (planar) lattice point theory consists in counting lattice points under the graph of positive functions supported on [0,M] and with radius of curvature comparable to M. We prove that, in some sense motivated by Feynman path integral formulation of Quantum Mechanics, for “most” functions the lattice error term in the area approximation is O(M1/2+ε). This complements Jarnı́k construction of curves with an optimal O(M2/3) error term.

Two problems associated with convex finite type domains

We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp $L^p$ estimates for $p>4$, generalizing the Carleson-Sjolin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.

A sum formula related to ellipsoids with applications to lattice point theory

In this article we consider finite and infinitep-dimensional sums over functionsf, where the argument off is represented by a positive definite quadratic form. We develop a sum formula like theEuler-Maclaurin orPoisson sum formula. Applications to exponential sums and lattice point problems are given.

Randomness in lattice point problems

Randomness in lattice point problems

On the distribution of Farey fractions and hyperbolic lattice points

We derive an asymptotic formula for the number of pairs of consecutive fractions a0=q0 and a=q in the Farey sequence of order Q such that a=q; q=Q; and (Q - q0)=q lie each in prescribed subintervals of the interval [0; 1]. We deduce the leading term in the asymptotic formula for `the hyperbolic lattice point problem" for the modular group PSL(2; Z ), the number of images of a given point under the action of the group in a given circle in the hyperbolic plane.

On logarithmically small errors in the lattice point problem

In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.

A lattice point problem associated with two polynomials

A lattice point problem associated with two polynomials

Lattice Point Problems and Distribution of Values of Quadratic Forms

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(rd-2). The estimate refines an earlier authors' bound of order O(rd-2) which holds for arbitrary ellipsoids, and is optimal for rational ellipsoids. As an application we prove a conjecture of Davenport and Lewis that the gaps between successive values, say s 0 as s - oo, for d > 9. For comparison note that sups(n(s) - s) 0, for rational Q[x] and d > 5. As a corollary we derive Oppenheim's conjecture for indefinite irrational quadratic forms, i.e., the set Q[Zd] is dense in R, for d > 9, which was proved for d > 3 by G. Margulis [Marl] in 1986 using other methods. Finally, we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums.

A note on a problem of lattice points in a certain planar domain

A note on a problem of lattice points in a certain planar domain

The Reciprocity Law for Dedekind Sums via the Constant Ehrhart Coefficient

These sums appear in various branches of mathematics: number theory, algebraic geometry, and topology; they have consequently been studied extensively in various contexts. These include the quadratic reciprocity law [13], random number generators [12], group actions on complex manifolds [9], and lattice point problems ([14] or [5]). Dedekind was the first to show the following reciprocity law [3]:

Random fluctuations of convex domains and lattice points

In this paper, we examine a random version of the lattice point problem. LetH\mathcal Hdenote the class of all homogeneous functions inC2(Rn)C^2(\mathbb R^n)of degree one, positive away from the origin. LetΦ\Phibe a random element ofH\mathcal H, defined on probability space(Ω,F,P)(\Omega ,\mathcal F,P), and define\[FΦ(ω,⋅)(ξ)=∫{x:Φ(ω,x)≤1}e−i⟨x,ξ⟩dxF_{\Phi (\omega ,\cdot )}(\xi )=\int _{\{x\colon \Phi (\omega ,x)\le 1\}}e^{-i\langle x,\xi \rangle }dx\]forω∈Ω\omega \in \Omega. We prove that, ifE(|FΦ(ξ)|)≤C[ξ]n+12E(|F_\Phi (\xi )|)\le C[\xi ]^{\frac {n+1}{2}}, where[ξ]=1+|ξ|[\xi ]=1+|\xi |, then\[E(NΦ)(t)=Vtn+O(tn−2+2n+1)E(N_\Phi )(t)=Vt^n+O(t^{n-2+\frac {2}{n+1}})\]whereV=E(|{x:Φ(⋅,x)≤1}|)V=E(|\{x\colon \Phi (\cdot ,x)\le 1\}|), the expected volume. That is, on average,NΦ(t)=Vtn+O(tn−2+2n+1)N_\Phi (t)=Vt^n+O(t^{n-2+\frac {2}{n+1}}). We give explicit examples in which the Gaussian curvature of{x:Φ(ω,x)≤1}\{x\colon \Phi (\omega ,x)\le 1\}is small with high probability, and the estimateNΦ(t)=Vtn+O(tn−2+2n+1)N_\Phi (t)=Vt^n+O(t^{n-2+\frac {2}{n+1}})holds nevertheless.

Uniform Lattice Point Estimates for Co-Finite Fuchsian Groups

In this paper we obtain uniform estimates for the lattice point problem in the hyperbolic plane H under the assumption that the action is by a Fuchsian group Γ which is co-finite. We fix a point w from H and set Nt(z, w) equal to the number of translates of w by the group Γ which lie in a geodesic ball of radius t centred at a point z of H. The behaviour of Nt(z, w) is then examined when t is large and z is allowed to vary over H. We show that the finite quantity depends crucially on the point w, and indeed can become arbitrarily large with w. On the other hand, for the average of this quotient we derive the estimate as t → ∞, where the implied constant is an explicit function of w. In this formula, vol(F) is the hyperbolic volume of a Dirichletfundamental domain F for Γ, and ∣Γw∣ denotes the number of elements from Γ fixing w. This estimate is then combined with a recent sampling theorem of K. Seip to obtain an inequality which decides whether or not the orbit Γ . w forms a set of interpolation for a given weighted Bergman space in H. 1991 Mathematics Subject Classification: 11F72, 11P21, 30D35, 30E05, 30F35.

On the Gauss mean-value formula for class number

Abstract.In his masterworkDisquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written asfor every ∊ > 0, whereH(−n) is, in modern notation,h(−4n). We also consider the average ofh(−n) itself obtaining the same type of result.Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.

Ergodic theory on SL ( n ), diophantine approximations and anomalies in the lattice point problem

Ergodic theory on SL ( n ), diophantine approximations and anomalies in the lattice point problem