arXiv subjects

Sungchul Lee

Publications and source records attributed to Sungchul Lee.

6 recordsLinked to original sources

arXiv2026

The Minkowski grid has robustly many repeated distances

We show that there exists a constant $\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}^2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, \[ \max_{\lambda>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=\lambda\} \gtrsim \frac{|A|^2}{n^{1-\delta}}.\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erd\H{o}s unit distance problem and the Elekes-R\'onyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erd\H{o}s. The existence of a set satisfying property (1) confirms a conjecture of Erd\H{o}s from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

math.CO↗

arXiv2026

Sharpness of convolution bounds for measures

In this paper, we determine the optimal universal $L^p$-$L^q$ type sets for convolution operators $f\mapsto \mu*f$ associated with fractal measures $\mu\in \mathcal P_{\alpha,\beta}(\mathbb R^d)$, which denotes the class of compactly supported Borel probability measures satisfying the $\alpha$-Frostman condition \[ \mu(B(x,\rho)) \lesssim \rho^\alpha, \qquad x\in\mathbb R^d,\quad 0<\rho<1, \] and the $\beta/2$-Fourier decay condition \[ |\widehat{\mu}(\xi)| \lesssim |\xi|^{-\beta/2}, \qquad \xi\in\mathbb R^d. \] More precisely, we characterize the largest $L^p$-$L^q$ region that is forced solely by the Frostman and Fourier decay assumptions throughout the full admissible range of $(\alpha,\beta)$, with distinct optimal regions in the geometric and nongeometric regimes. We prove optimality in the worst-case sense over $\mathcal P_{\alpha,\beta}(\mathbb R^d)$ by constructing, for each admissible pair $(\alpha,\beta)$, a single extremal measure whose support has the smallest Hausdorff dimension allowed by the hypotheses. Moreover, variants of the same constructions also yield a single-measure sharpness theorem for the $L^2$ Fourier restriction theorem of Mockenhaupt--Mitsis--Bak--Seeger: in every dimension and in both the geometric and nongeometric regimes, we construct a measure in $\mathcal P_{\alpha,\beta}(\mathbb R^d)$ for which the Mockenhaupt--Mitsis--Bak--Seeger threshold exponent is sharp.

math.CA↗

arXiv2025

Forecasting VIX using interpretable Kolmogorov-Arnold networks

This paper presents the use of Kolmogorov-Arnold Networks (KANs) for forecasting the CBOE Volatility Index (VIX). Unlike traditional MLP-based neural networks that are often criticized for their black-box nature, KAN offers an interpretable approach via learnable spline-based activation functions and symbolification. Based on a parsimonious architecture with symbolic functions, KAN expresses a forecast of the VIX as a closed-form in terms of explanatory variables, and provide interpretable insights into key characteristics of the VIX, including mean reversion and the leverage effect. Through in-depth empirical analysis across multiple datasets and periods, we show that KANs achieve competitive forecasting performance while requiring significantly fewer parameters compared to MLP-based neural network models. Our findings demonstrate the capacity and potential of KAN as an interpretable financial time-series forecasting method.

cs.LG↗

arXiv2019

Deep learning for undersampled MRI reconstruction

This paper presents a deep learning method for faster magnetic resonance imaging (MRI) by reducing k-space data with sub-Nyquist sampling strategies and provides a rationale for why the proposed approach works well. Uniform subsampling is used in the time-consuming phase-encoding direction to capture high-resolution image information, while permitting the image-folding problem dictated by the Poisson summation formula. To deal with the localization uncertainty due to image folding, very few low-frequency k-space data are added. Training the deep learning net involves input and output images that are pairs of Fourier transforms of the subsampled and fully sampled k-space data. Numerous experiments show the remarkable performance of the proposed method; only 29% of k-space data can generate images of high quality as effectively as standard MRI reconstruction with fully sampled data.

stat.ML↗

arXiv2006

Entropy method for the left tail

When we use the entropy method to get the tail bounds, typically the left tail bounds are not good comparing with the right ones. Up to now this asymmetry has been observed many times. Surprisingly we find an entropy method for the left tail that works in the exactly same way that it works for the right tail.

math.PR↗

arXiv2006

Rates of convergence of means of Euclidean functionals

Let $L$ be the Euclidean functional with $p$-th power-weighted edges. Examples include the sum of the $p$-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (1994, 1996) have shown that for $n$ i.i.d. sample points $\{X_1,...,X_n\}$ from $[0,1]^d$, $L(\{X_1,...,X_n\})/n^{(d-p)/d}$ converges a.s. to a finite constant. Here we bound the rate of convergence of $EL(\{X_1,...,X_n\})/n^{(d-p)/d}$.

math.PR↗