Finite Point Configurations and the Regular Value Theorem in a Fractal setting

The failure of the fractal uncertainty principle for the Walsh--Fourier transform

We construct $δ$-regular sets with $δ\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.

分散型Furstenberg集II：一般构型与分形限制投影

Spread-out Furstenberg sets II: general configurations and fractal restricted projections

该论文针对阴影未局部集中于 pencil 的特定构型证明最优对偶 Furstenberg 估计，推导平面及高维限制投影定理，并将其应用于 Grassmann 流形曲面投影、固定距离等多个分形相关问题。

Patterns in Random Fractals

We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemerédi theorems for random discrete sets, we also consider the corresponding problem for sets of positive $ν$-measure, where $ν$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating t…

Fourier integral operators, fractal sets and the regular value theorem

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{\mathcal H}({\mathcal E})$, and $ϕ$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: ϕ_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{\mathcal H}({\mathcal E})-m$, 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 …

Dolgopyat's method and the fractal uncertainty principle

We show a fractal uncertainty principle with exponent $1/2-δ+ε$, $ε>0$, for Ahflors-David regular subsets of $\mathbb R$ of dimension $δ\in (0,1)$. This improves over the volume bound $1/2-δ$, and $ε$ is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Here the group invariance of the set is replaced by its fractal structure. As an application, we quantify the result of Naud on spectral gaps for convex co-compact hyperbolic surfaces and o…

A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension

Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - ε_d < \dim_H(A) < 1$. Then for any polynomial $P$ of degree $d$ with no constant term, there exists a point configuration $\{ x, x-t,x-P(t) \} \subset A$ with $t \approx_P 1$.

傅里叶-贝塞尔变换的分形不确定性与定量唯一性

Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

针对傅里叶-贝塞尔变换，研究人员建立了其定量唯一性与分形不确定性原理，通过适配正则集合的阻尼函数及多尺度迭代等方法，克服了经典傅里叶论证无法直接应用的问题，得到了相关函数估计式。

Multilinear generalized Radon transforms and point configurations

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, with $k \ge 2$, including the distribution of simplices, volumes and angles determined by the points of fractal subsets $E \subset {\Bbb R}^d$, $d \ge 2$. If $T_k(E)$ denotes the set of noncongruent $(k+1)$-point configurations determined by $E$, we show that if the Hausdorff dimension of $E$ is greater than $d-\frac{d-1}{2k}$, then th…

随机Cantor集的尖锐Favard长度

Sharp Favard length of random Cantor sets

研究平面一维随机分形邻域的Favard长度，证明其与log^{-1}(1/r)同阶，并揭示不同随机分形中衰减率的多样性。

高维情形下借助Dolgopyat方法的分形不确定性原理

The fractal uncertainty principle via Dolgopyat's method in higher dimensions

该研究将Dolgopyat方法推广至高维情形，证明了满足非正交性条件的Ahlfors-David正则子集的分形不确定性原理，并得到任意维凸紧双曲流形上拉普拉斯算子的定量谱隙。