Distribution Of Lattice Points Research Articles

Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles

AbstractHardy conjectured that the error term arising from approximating the number of lattice points lying in a radius‐ disc by its area is . One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

AbstractWe asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a “microsquare”, a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density 1. subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume with .We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weighting functions for annuli and for balls. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

О научных трудах Дмитрия Александровича Попова

Настоящий очерк посвящён рассказу о жизни и научном творчестве замечательного российского математика Дмитрия Александровича Попова, отметившего свой 80-летний юбилей в августе 2019 г. Д.А. Попов внёс значительный вклад в математические основы рентгеновской, ультразвуковой и акустической томографии, теорию оценок осциллирующих интегралов, в задачи, связанные с распределением целых точек в круге на плоскостии в телах вращения. Особое место в очерке отводится недавним результатам юбиляра по проблемам связи спектра оператора Лапласа на фундаментальной области модулярной группы с распределением простых чисел.

Lattice point problems involving index and joint visibility

We study questions concerning the distribution of lattice points in dimensions two and higher. We give asymptotic formulas for the number of integer lattice points of fixed index visible from certain admissible sets. We also study the shape of the body P A 1 , … , P A k PA_1, \dots , PA_k when the lattice points A 1 , … , A k A_1, \dots , A_k move inside a given large ball, and P P is a lattice point visible from all A i A_i ’s.

Fourier integral operators, fractal sets, and the regular value theorem

We prove that if E ⊂ R 2 d , for d ⩾ 2 , is an Ahlfors–David regular product set of sufficiently large Hausdorff dimension, denoted by dim H ( E ) , and ϕ is a sufficiently regular function, then the upper Minkowski dimension of the set { w ∈ E : ϕ l ( w ) = t l ; 1 ⩽ l ⩽ m } does not exceed dim H ( E ) − m , in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.

Distribution of Lattice Points

We discuss the lattice structure of congruential random number generators and examine figures of merit. Distribution properties of lattice measures in various dimensions are demonstrated by using large numerical data. Systematic search methods are introduced to diagnose multiplier areas exhibiting good, bad and worst lattice structures. We present two formulae to express multipliers producing worst and bad laice points. The conventional criterion of normalised lattice rule is also questioned and it is shown that this measure used with a fixed threshold is not suitable for an effective discrimination of lattice structures. Usage of percentiles represents different dimensions in a fair fashion and provides consistency for different figures of merits.

On the distribution of lattice points in thin annuli

We show that the number of lattice points lying in a thin annulus has a Gaussian value distribution if the width of the annulus tends to zero sufficiently slowly as we increase the inner radius.

Distribution of lattice points on surfaces of second order

Let f1(x1,..., xl1) and f2(y1,...,yl2) be positive definite primitive quadratic forms in l1 and l2 variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone f1(x1,...,xl1)=f2(y1,...,yl2), in the domain f1(x1,...,xl1)≦N for N»∞. Our technical tool is the Rankin-Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in l variables, l odd, l≧5 are obtained.

Distribution of lattice points on hyperbolic surfaces

Let two lattices $Λ', Λ'' ⊂ ℝ^s$ have the same number of points on each hyperbolic surface $|x₁...x_s| = C$. We investigate the case when Λ', Λ'' are sublattices of $ℤ^s$ of the same prime index and show that then Λ' and Λ'' must coincide up to renumberin