Mean lattice point discrepancy bounds, II: Convex domains in the plane

Bounding regions to plane steepest descent curves of quasi convex families

Two dimensional steepest descent curves (SDC) for a quasi convex family are considered; the problem of their extensions (with constraints) outside of a convex body $K$ is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on $K$. These regions are bounded by arcs of involutes of the boundary of $K$ and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self contracting sets (with opp…

$L^p$ norms of the lattice point discrepancy

We estimate the $L^{p}$ norms of the discrepancy between the volume and the number of integer points in $rΩ-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $Ω$ in $\mathbb{R}^{d}$ with smooth boundary with strictly positive curvature, \[ \left\{ {\displaystyle\int_{\mathbb R}}{\displaystyle\int_{\mathbb{T}^{d}}}\left\vert \sum_{k\in\mathbb{Z}^{d}}χ_{rΩ-x}(k)-r^{d}\left\vert Ω\right\vert \right\vert ^{p}dxdμ(r-R) \right\} ^{1/p}, \] where $μ$ is a Borel measure compactly supported on the positive real axis and $R\to+\infty$.

Fuglede conjecture holds for convex planar domains

Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.

偏差与具有主导混合光滑性的函数空间中的积分

Discrepancy and integration in function spaces with dominating mixed smoothness

本文研究了具有主导混合光滑性的函数空间中偏差的最优下界及积分误差，通过分析点集并引入Chen和Skriganov的构造，推导了任意维度下的最优上界及Triebel-Lizorkin和Sobolev空间中的结果。

The Spherical Cap Discrepancy of HEALPix Points

In this paper we show that the spherical cap discrepancy of the point set given by centers of pixels in the HEALPix tessellation (short for Hierarchical, Equal Area and iso-Latitude Pixelation) of unit $2$-sphere is lower and upper bounded by order square root of the number of points, and compute explicit constants. This adds to the known collection of explicitly constructed sets whose discrepancy converges with order $N^{-1/2}$, matching the asymptotic order for i.i.d. random point sets. We describe the HEALPix framework in more detail and give explicit formulas for the boundaries and pixel c…

平面中与粗糙凸区域相关的 Bochner–Riesz 算子的改进 $L^p$ 界

Improved $L^p$ bounds for Bochner--Riesz operators associated with rough convex domains in the plane

本文利用 Bose–Chowla 的大 $B_m$ 集与 Bourgain 的大 $\Lambda(p)$ 集构造平面凸区域，改进了广义 Bochner–Riesz 算子的 $L^p$ 估计。

平面上简单非凸域上的Bellman函数

Bellman functions on simple non-convex domains in the plane

该研究推广了此前BMO上积分泛函的Bellman函数相关工作，提出构造两无界凸域之差构成的平面域上最小局部凹函数的算法，可得到多类函数的精确估计。

平面凸体的平衡态

Equilibria of plane convex bodies

本研究推导了平面凸体相对于质心的水平平衡态数量公式，该公式以凸体边界渐屈线的环绕数表示，可推广至质心在渐屈线上的情形，且修正版适用于非水平平衡态。

Right order Turan-type converse Markov inequalities for convex domains on the plane

For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Turán, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although parti…

Convex-concave body in $\mathbb{R}P^3$ contains a line

We define a class of $L$-convex-concave subsets of $\mathbb{R}P^3$, where $L$ is a projective line in $\mathbb{R}P^3$. These are sets whose sections by any plane containing $L$ are convex and concavely depend on this plane. We prove a version of Arnold hypothesis for these sets, namely we prove that each such set contains a line.