Variance of lattice point counting in some special shells in $\mathbb{R}^d$

三维收缩 Cygan-Korányi 球壳的格点计数统计

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

本文研究三维收缩 Cygan-Korányi 球壳中的整数格点计数误差，证明归一化误差在间隙宽度趋于零且正则时具有由矩确定的极限分布，并刻画高斯与非高斯情形。

A lower bound for the variance in arithmetic progressions of some multiplicative functions close to $1$

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $α_N$-fold divisor functions, when $α_N$ is a sequence of positive real numbers approaching $1$ in a suitable way or $α_N=1$, and the indicator of $y$-smooth numbers, for suitably large parameters $y$. As a corollary, we will strengthen a previous author's result on the first subject and obtain matching lower bounds to some Barban-Davenport-Halberstam type theorems for $y$-smooth numbers. Incidentally, we will also find a lower bound for the vari…

大维度中的双曲格点计数问题

The hyperbolic lattice counting problem in large dimensions

本文研究大维度双曲空间中余紧格作用下的圆问题误差项平均行为，给出局部平均上界并证明均值与二阶矩的Ω结果。

Lattice deformations in the Heisenberg group

The space of deformations of the integer Heisenberg group under the action of $\textrm{Aut}(H(\mathbb{R}))$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and variance estimates for Heisenberg lattice point counting in measurable subsets of $\mathbb{R}^3$; in particular, we obtain a random Minkowski-type theorem. Unlike the Euclidean case, we show there are necessary geometric conditions on the sets that satisfy effective variance bounds.

双曲球面问题中的局部平均

Local average in the Hyperbolic sphere problem

本文研究了双曲球面问题中的局部平均，通过改进余项界和利用谱指数和的界，改进了双曲格点计数的精度。