arXiv subjects

Yoav A. Gath

Publications and source records attributed to Yoav A. Gath.

5 recordsLinked to original sources

arXiv2024

Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells

Let $E(x;ω)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Korányi spherical shell of inner radius $x$ and gap width $ω(x)>0$. Assuming that $ω(x)\to0$ as $x\to\infty$, and that $ω$ satisfies suitable regularity conditions, we prove that $E(x;ω)$, properly normalized, has a limiting distribution. Moreover, we show that the corresponding distribution is moment-determinate, and we give a closed form expression for its moments. As a corollary, we deduce that the limiting distribution is the standard Gaussian measure whenever $ω$ is slowly varying. We also construct gap width functions $ω$, whose corresponding error term has a limiting distribution that is absolutely continuous with a non-Gaussian density.

math.NT↗

arXiv2021

Distribution and Moments of the Error Term in the Lattice Point Counting Problem for 3-Dimensional Cygan-Korányi Balls

We study fluctuations of the error term for the number of integer lattice points lying inside a 3-dimensional Cygan-Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.

math.NT↗

arXiv2020

On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Kor{á}nyi norm ball of large radius. Let $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\capδ_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$ denote the error term which occurs for this lattice point counting problem on the Heisenberg group $\mathbb{H}_{q}$, where $\mathcal{B}$ is the unit ball in the Cygan-Kor{á}nyi norm and $δ_{x}$ is the Heisenberg-dilation by $x>0$. For $q\geq3$ we consider the suitably normalized error term $\mathcal{E}_{q}(x)/x^{2q-1}$, and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by $\mathcal{P}_{q}(α)$, can be extended to the whole complex plane $\mathbb{C}$ as an entire function of $α$ and satisfies for any non-negative integer $j\geq0$ and any $α\in\mathbb{R}$, $|α|>α_{q,j}$, the bound: \begin{equation*} \begin{split} \big|\mathcal{P}^{(j)}_{q}(α)\big|\leq\exp{\Big(-|α|^{4-β/\log\log{|α|}}\Big)} {split} {equation*} where $β>0$ is an absolute constant. In addition, we give an explicit formula for the $j$-th integral moment of the density $\mathcal{P}_{q}(α)$ for any integer $j\geq1$.

math.NT↗

arXiv2019

On an Analogue Of the Gauss Circle Problem For the Heisenberg Groups

We consider the problem of estimating the error term $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\capδ_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$ which occurs in the counting of lattice points in Heisenberg dilates of the Cygan-Kor{á}nyi ball. We prove three type of results regarding the order of magnitude of $\mathcal{E}_{q}(x)$, which are valid for any $q\geq3$. An upper bound estimate of the form $|\mathcal{E}_{q}(x)|\ll x^{2q-2/3}$ ; A sharp second moment estimate, which shows that $\mathcal{E}_{q}(x)$ has order of magnitude $x^{2q-1}$ in mean-square ; And an $Ω$-estimate of the form $\mathcal{E}_{q}(x)=Ω\big(x^{2q-1}\big(\log{x}\big)^{1/4}\big(\log{\log{x}}\big)^{1/8}\big)$. Consequently, we obtain the lower bound $κ_{q}=\sup\big\{α>0:\big|\mathcal{E}_{q}(x)\big|\ll x^{2q+2-α}\big\}\geq\frac{8}{3}$ for $q\geq3$, and conjecture that $κ_{q}=3$

math.NT↗

arXiv2017

On the best possible exponent for the error term in the lattice point counting problem on the first Heisenberg group

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 $\mathcal{N}_{4,1}(z,w)=(|z|^{4}+w^{2})^{1/4}$. In this case, our main theorem establishes the estimate $\mathcal{E}(x)=Ω_{\pm}(x^{\frac{1}{2}})$, where $\mathcal{E}(x)=\mathcal{S}(x)-\frac{π^{2}}{2}x$ is the error term arising in the lattice point counting problem, $\mathcal{S}(x)$ is given by $$\mathcal{S}(x)=\sum_{0\leq m^2+n^2<\, x}r_2(m)$$ and $r_2(m)=|\{a,b\in\mathbb{Z}:\,a^{2}+b^{2}=m\}|$ is the familiar sum of squares function. As a corollary, we deduce that the exponent $\frac{1}{2}$ in the upper-bound $\left|\mathcal{E}(x)\right|\ll x^{\frac{1}{2}}\log{x}$ obtained by Garg, Nevo & Taylor can not be improved and is thus best possible, thereby resolving the lattice point counting problem for the case in hand.

math.NT↗