arXiv subjects

Bochen Liu

Publications and source records attributed to Bochen Liu.

27 records · Page 2Linked to original sources

arXiv2019

On Hausdorff dimension of radial projections

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $π^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$π^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{R}^d$ the radial projection $π^x$ preserves the Hausdorff dimension of $E$, namely whether $\dim_{\mathcal{H}}π^x(E)=\dim_{\mathcal{H}} E$. We develop a general framework to link $π^x(E)$, $x\in F$ and $π^y(F)$, $y\in E$, for any Borel set $F\subset\mathbb{R}^d$. In particular, whether $\dim_{\mathcal{H}}π^x(E)=\dim_{\mathcal{H}}E$ for some $x\in F$ can be reduced to whether $F$ is visible from some $y\in E$ (i.e. $\mathcal{H}^{d-1}(π^y(F))>0$). This allows us to apply Orponen's estimate on visibility to obtain $$\dim_{\mathcal{H}}\left\{x\in\mathbb{R}^d: \dim_{\mathcal{H}}π^x(E)<\dim_{\mathcal{H}}E\right\}\leq 2(d-1)-\dim_{\mathcal{H}}E,$$ for any Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\in(d-2, d-1]$. This improves the Peres-Schlag bound when $\dim_{\mathcal{H}} E\in(d-\frac{3}{2}, d-1]$, and it is optimal at the endpoint $\dim_{\mathcal{H}} E=d-1$.

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arXiv2018

An $L^2$-identity and pinned distance problem

Let $μ$ be a Frostman measure on $E\subset\mathbb{R}^d$. The spherical average decay $$\int_{S^{d-1}}|\widehatμ(rω)|^2\,dω\lesssim r^{-β} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly. The idea is to reduce the pinned distance problem to an integral where spherical averages apply. The key ingredient is the following identity. Using a group action argument, we show that for any Schwartz function $f$ on $\mathbb{R}^d$ and any $x\in\mathbb{R}^d$, $$\int_0^\infty |ω_t*f(x)|^2\,t^{d-1}dt\,=\int_0^\infty|\widehat{ω_r}*f(x)|^2\,r^{d-1}dr,$$ where $ω_r$ is the normalized surface measure on $r S^{d-1}$. An interesting remark is that the right hand side can be easily seen equal to $$c_d\int\left|D_x^{-\frac{d-1}{2}}e^{-2πi t\sqrt{-Δ}}f(x)\right|^2\,dt=c_d'\int\left|D_x^{-\frac{d-2}{2}}e^{2πi tΔ}f(x)\right|^2\,dt.$$ An alternative derivation of Mattila's integral via group actions is also given in the Appendix.

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arXiv2018

Group actions, the Mattila integral and applications

The Mattila integral, $$ {\mathcal M}(μ)=\int {\left( \int_{S^{d-1}} {|\widehatμ(r ω)|}^2 dω\right)}^2 r^{d-1} dr,$$ developed by Mattila, is the main tool in the study of the Falconer distance problem. In this paper, with a very simple argument, we develop a generalized version of the Mattila integral. Our first application is to consider the product of distances $$(Δ(E))^k= \left\{\prod_{j=1}^k |x^j-y^j|: x^j, y^j\in E\right\} $$ and show that when $d\geq 2$, $(Δ(E))^k$ has positive Lebesgue measure if $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4k-1}$. Another application is, we prove for any $E,F,H\subset\mathbb{R}^2$, $\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)+\dim_{\mathcal{H}}(H)>4$, the set $$E\cdot(F+H)=\{x\cdot(y+z): x\in E, y\in F, z\in H\}$$ has positive Lebesgue.

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arXiv2017

Improvement on $2$-chains inside thin subsets of Euclidean spaces

We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$Δ_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has positive Lebesgue measure. It generalizes Wolff-Erdogan's result on distances and improves a result of Bennett, Iosevich and Taylor on finite chains. We also consider the similarity class of $2$-chains, $$S_2(E)=\left\{\frac{t_1}{t_2}:(t_1,t_2)\inΔ_2(E)\right\}=\left\{\frac{|x-y|}{|y-z|}: x, y, z\in E \right\}\subset\mathbb{R},$$ and show that $|S_2(E)|>0$ whenever $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{7}$.

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arXiv2017

Pinned distance problem, slicing measures and local smoothing estimates

We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$Δ^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a probability measure $μ_F$ on $F$ such that for $μ_F$-a.e. $y\in F$, (1) $\dim_{\mathcal H}(Δ^y(E))\geqβ$ if $\dim_{\mathcal H}(E) + \frac{d-1}{d+1}\dim_{\mathcal H}(F) > d - 1 + β$; (2) $Δ^y(E)$ has positive Lebesgue measure if $\dim_{\mathcal H}(E)+\frac{d-1}{d+1}\dim_{\mathcal H}(F) > d$; (3) $Δ^y(E)$ has non-empty interior if $\dim_{\mathcal H}(E)+\frac{d-1}{d+1}\dim_{\mathcal H}(F) > d+1$. We also show that in the case when $\dim_{\mathcal H}(E)+\frac{d-1}{d+1}\dim_{\mathcal H}(F)>d$, for $μ_F$-a.e. $y\in F$, $$ \left\{t\in{\Bbb R} : \dim_{\mathcal H}(\{x\in E:|x-y|=t\}) \geq \dim_{\mathcal H}(E)+\frac{d+1}{d-1}\dim_{\mathcal H}(F)-d \right\} $$ has positive Lebesgue measure. This describes dimensions of slicing subsets of $E$, sliced by spheres centered at $y$. In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.

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arXiv2016

An elementary approach to simplexes in thin subsets of Euclidean space

We prove that if the Hausdorff dimension of $E \subset {\Bbb R}^d$, $d \ge 3$, is greater than $\min \left\{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right\},$ then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$, the set of congruence classes of $k$-dimensional simplexes with vertices in $E$, is positive. This improves the best bounds previously known, decreasing the $\frac{d+k+1}{2}$ threshold obtained in Erdoğan-Hart-Iosevich (2012) to $\frac{d+k}{2}$ via a different and conceptually simpler method. We also give a simpler proof of the $d-\frac{d-1}{2d}$ threshold for $d$-dimensional simplexes obtained in Greenleaf-Iosevich (2012), Grafakos-Greenleaf-Iosevich-Palsson (2015).

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arXiv2016

Equilateral triangles in subsets of ${\Bbb R}^d$ of large Hausdorff dimension

We prove that subsets of ${\Bbb R}^d$, $d \ge 4$ of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two dimensions (see \cite{CLP14}). We show that no extra conditions are needed in dimensions four and higher. The three dimensional case remains open. Some interesting parallels exist between the triangle problem in Euclidean space and its counter-part in vector spaces over finite fields. We shall outline these similarities in hopes of eventually achieving a comprehensive understanding of this phenomenon in the setting of locally compact abelian groups.

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arXiv2015

Falconer distance problem, additive energy and Cartesian products

A celebrated result due to Wolff says if $E$ is a compact subset of ${\Bbb R}^2$, then the Lebesgue measure of the distance set $Δ(E)=\{|x-y|: x,y \in E \}$ is positive if the Hausdorff dimension of $E$ is greater than $\frac{4}{3}$. In this paper we improve the $\frac{4}{3}$ barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's $\frac{d}{2}+\frac{1}{3}$ exponent to $\frac{d^2}{2d-1}$. The proof uses a combination of Fourier analysis and additive comibinatorics.

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arXiv2013

On radii of spheres determined by subsets of Euclidean space

In this paper we consider the problem of how large the Hausdorff dimension of $E\subset\R^d$ needs to be in order to ensure that the radii set of $(d-1)$-dimensional spheres determined by $E$ has positive Lebesgue measure. We also study the question of how often can a neighborhood of a given radius repeat. We obtain two results. First, by applying a general mechanism developed in \cite{mul} for studying Falconer-type problems, we prove that a neighborhood of a given radius cannot repeat more often than the statistical bound if $\dH(E)>d-1+\frac{1}{d}$; In $\R^2$, the dimensional threshold is sharp. Second, by proving an intersection theorem, we prove for a.e $a\in\R^d$, the radii set of $(d-1)$-spheres with center $a$ determined by $E$ must have positive Lebesgue measure if $\dH(E)>d-1$, which is a sharp bound for this problem.

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