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Shukun Wu

Publications and source records attributed to Shukun Wu.

32 records · Page 2Linked to original sources
arXiv2023

A note on the largest sum-free sets of integers

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+ω(N)$ for some increasing unbounded function $ω$. The question is generally attacked in the literature by considering another conjecture, which asserts that as $N\to\infty$, $\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}({\bf 1}_{(1/3,2/3)}-1/3)(nx)\to\infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\geq1$. In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set $A$, which might be of independent interest.

math.CO↗
arXiv2023

Channel and Spatial Relation-Propagation Network for RGB-Thermal Semantic Segmentation

RGB-Thermal (RGB-T) semantic segmentation has shown great potential in handling low-light conditions where RGB-based segmentation is hindered by poor RGB imaging quality. The key to RGB-T semantic segmentation is to effectively leverage the complementarity nature of RGB and thermal images. Most existing algorithms fuse RGB and thermal information in feature space via concatenation, element-wise summation, or attention operations in either unidirectional enhancement or bidirectional aggregation manners. However, they usually overlook the modality gap between RGB and thermal images during feature fusion, resulting in modality-specific information from one modality contaminating the other. In this paper, we propose a Channel and Spatial Relation-Propagation Network (CSRPNet) for RGB-T semantic segmentation, which propagates only modality-shared information across different modalities and alleviates the modality-specific information contamination issue. Our CSRPNet first performs relation-propagation in channel and spatial dimensions to capture the modality-shared features from the RGB and thermal features. CSRPNet then aggregates the modality-shared features captured from one modality with the input feature from the other modality to enhance the input feature without the contamination issue. While being fused together, the enhanced RGB and thermal features will be also fed into the subsequent RGB or thermal feature extraction layers for interactive feature fusion, respectively. We also introduce a dual-path cascaded feature refinement module that aggregates multi-layer features to produce two refined features for semantic and boundary prediction. Extensive experimental results demonstrate that CSRPNet performs favorably against state-of-the-art algorithms.

cs.CV↗
arXiv2023

Kakeya sets from lines in $SL_2$

We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was also recently proved by Fässler and Orponen using different techniques. Our method combines induction on scales with a special structural property of $SL_2$ Kakeya sets, which says that locally such sets look like the pre-image of an arrangement of plane curves above a special type of map from $\mathbb{R}^3$ to $\mathbb{R}^2$, called a twisting projection. This reduces the study of $SL_2$ Kakeya sets to a Kakeya-type problem for plane curves; the latter is analyzed using a variant of Wolff's circular maximal function.

math.CA↗
arXiv2023

Control Input Inference of Mobile Agents under Unknown Objective

Trajectory and control secrecy is an important issue in robotics security. This paper proposes a novel algorithm for the control input inference of a mobile agent without knowing its control objective. Specifically, the algorithm first estimates the target state by applying external perturbations. Then we identify the objective function based on the inverse optimal control, providing the well-posedness proof and the identifiability analysis. Next, we obtain the optimal estimate of the control horizon using binary search. Finally, the agent's control optimization problem is reconstructed and solved to predict its input. Simulation illustrates the efficiency and the performance of the algorithm.

cs.RO↗
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↗
arXiv2021

A note on the refined Strichartz estimates and maximal extension operator

There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain-Demeter's $l^2$ decoupling theorem and induction on scales. In the second part, we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main methods are sharp $L^2$ estimates by Du and Zhang, and the bilinear method by Wolff and Tao.

math.CA↗
arXiv2021

The Bochner-Riesz problem: an old approach revisited

We show that the recent techniques developed to study the Fourier restriction problem apply equally well to the Bochner-Riesz problem. This is achieved via applying a pseudo-conformal transformation and a two-parameter induction-on-scales argument. As a consequence, we improve the Bochner-Riesz problem to the best known range of the Fourier restriction problem in all high dimensions.

math.CA↗
arXiv2021

The largest $(k, \ell)$-sum-free subsets

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $ω(N)\to\infty$ as $N\to\infty$, such that $cN+ω(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $ω(N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell)$-sum-free sets and restricted $(k,\ell)$-sum-free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$-sum-free sets, and confirm the conjecture for infinitely many $(k,\ell)$.

math.CO↗