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Shengwen Gan

Publications and source records attributed to Shengwen Gan.

26 records · Page 2Linked to original sources
arXiv2023

Exceptional set estimates for radial projections in $\mathbb{R}^n$

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{dim}(\{y\in \mathbb{R}^n \setminus A\mid \text{dim} (π_y(A)) < s\})\leq \max\{k+s -\dim A,0\}. \] The second conjecture is by Liu: Given a Borel set $A\subset \mathbb{R}^n$, then \[ \text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(π_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil. \]

math.CA↗
arXiv2022

An exceptional set estimate for restricted projections to lines in $\mathbb{R}^3$

Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ''(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $l_θ=\{tγ(θ):t\in\mathbb{R}\}$ and $ρ_θ:\mathbb{R}^3\rightarrow l_θ$ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $A\subset\mathbb{R}^3$ and $0\le s\le 1$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(ρ_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+\frac{s-\text{dim}(A)}{2}$.

math.CA↗
arXiv2022

Square function estimates for conical regions

We prove square function estimates for certain conical regions. Specifically, let $\{Δ_j\}$ be regions of the unit sphere $\mathbb{S}^{n-1}$ and let $S_j f$ be the smooth Fourier restriction of $f$ to the conical region $\{ξ\in\mathbb{R}^n:ξ/|ξ|\inΔ_j\}$. We are interested in the following estimate $$\Big\|(\sum_j|S_jf|^2)^{1/2}\Big\|_p\lesssim_εδ^{-ε}\|f\|_p.$$ The first result is: when $\{Δ_j\}$ is a set of disjoint $δ$-balls, then the estimate holds for $p=4$. The second result is: In $\mathbb{R}^3$, when $\{Δ_j\}$ is a set of disjoint $δ\timesδ^{1/2}$-rectangles contained in the band $\mathbb{S}^2\cap N_δ(\{ξ_1^2+ξ_2^2=ξ_3^2\})$ and ${\rm{supp}}\widehat f\subset \{ξ\in\mathbb{R}^3:ξ/|ξ|\in\mathbb{S}^2\cap N_δ(\{ξ_1^2+ξ_2^2=ξ_3^2\})\}$, then the estimate holds for $p=8$. The two estimates are sharp.

math.CA↗
arXiv2020

A Sum-Product Estimate for Well Spaced Sets

We study the $δ$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $α\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $δ=|A|^{-α}$, we have: $\mathcal{N}(A+A, δ)\cdot \mathcal{N}(AA, δ) \gtrsim_ε|A|δ^{-1+ε}$.

math.CO↗