Spectral and Tiling properties of the Unit Cube

单位质量晶体型测度的一个新例子及其在函数平移对实数直线铺砌中的应用

A new example of crystalline-type measure with unit masses and an application to tiling of the real line by translates of a function

本文构造了密度无界的离散集Λ，其计数测度的傅里叶变换为含δ₀''项的缓增分布，并用此构造出具有紧支撑函数对实数直线的无界密度平移铺砌。

从单位立方体到某些对称空间的保测映射

Measure-preserving mappings from the unit cube to some symmetric spaces

该研究构造了从d维单位立方体到d维单位球及各类紧秩一对称空间的保测映射，还给出了特定条件下生成到乘积空间与纤维丛的保测映射的方法。

用函数平铺p进数域$\n\mathbb{Q}_p$

Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function

本研究探讨p进数域中可平移平铺函数的性质，证明其一致局部常值性，解答Leptin与Müller的问题并给出离散集对应一致单位分解的充要条件，关联Fuglede猜想，刻画乘积空间平铺集结构并证明其为谱集。

On $k$-antichains in the unit $n$-cube

A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \[ \text{card}(A \cap C) \le k, \, \text{ for all chains } \, C \subset [0,1]^n \, , \] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the b…

A Szemerédi-type theorem for subsets of the unit cube

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit…

Quantitative bounds for product of simplices in subsets of the unit cube

For each $1\leq i \le n$, let $k_i\geq 1$ and let $Δ_i$ be a set of vertices of a non-degenerate simplex of $k_i+1$ points in $\mathbb{R}^{k_i+1}$. If $A\subseteq [0,1]^{k_1+1}\times \cdots \times [0,1]^{k_n+1}$ is a Lebesgue measurable set of measure at least $δ$, we show that there exists an interval $I=I(Δ_1,\ldots, Δ_n,A)$ of length at least $\exp(-δ^{-C(Δ_1,\ldots, Δ_n)})$ such that for each $λ\in I$, the set $A$ contains $Δ'_1\times \cdots \times Δ'_n$, where each $Δ_i'$ is an isometric copy of $λΔ_i$. This is a quantitative improvement of a result by Lyall and Magyar. Our proof relies o…

On the structure of spectral and tiling subsets of cyclic groups

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb{R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structu…