arXiv subjects

Dingding Yu

Publications and source records attributed to Dingding Yu.

5 recordsLinked to original sources

arXiv2026

On the maximal run-length function in the L\"uroth expansion

Let $ \ell_n(x) $ denote the maximal run-length among the first $ n $ digits of the L\"{u}roth expansion of $ x\in(0,1] $. While $ \ell_n(x) $ grows logarithmically, we investigate the finer multifractal properties of the exceptional set where $\ell_n(x)$ exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set \[ \left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = \alpha, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = \beta \right\}, \] for all $ 0 \le \alpha \le \beta \le 1 $.

math.MG↗

arXiv2025

EmoHead: Emotional Talking Head via Manipulating Semantic Expression Parameters

Generating emotion-specific talking head videos from audio input is an important and complex challenge for human-machine interaction. However, emotion is highly abstract concept with ambiguous boundaries, and it necessitates disentangled expression parameters to generate emotionally expressive talking head videos. In this work, we present EmoHead to synthesize talking head videos via semantic expression parameters. To predict expression parameter for arbitrary audio input, we apply an audio-expression module that can be specified by an emotion tag. This module aims to enhance correlation from audio input across various emotions. Furthermore, we leverage pre-trained hyperplane to refine facial movements by probing along the vertical direction. Finally, the refined expression parameters regularize neural radiance fields and facilitate the emotion-consistent generation of talking head videos. Experimental results demonstrate that semantic expression parameters lead to better reconstruction quality and controllability.

cs.CV↗

arXiv2025

Multifractal analysis of maximal product of consecutive partial quotients in continued fractions

Let $[a_1(x), a_2(x), \ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. We study the growth rate of the maximal product of consecutive partial quotients among the first $n$ terms, defined by $L_n(x)=\max_{1\leq i\leq n}\{a_i(x)a_{i+1}(x)\}$, from the viewpoint of multifractal analysis. More precisely, we determine the Hausdorff dimension of the level set \[L(\varphi):=\left\{x\in (0,1):\lim_{n\to \infty}\frac{L_n(x)}{\varphi(n)}=1\right\},\] where $\varphi:\mathbb{R^+}\to\mathbb{R^+}$ is an increasing function such that $\log \varphi$ is a regularly increasing function with index $\rho$. We show that there exists a jump of the Hausdorff dimension of $L(\varphi)$ when $\rho=1/2$. We also construct uncountably many discontinuous functions $\psi$ that cause the Hausdorff dimension of $L(\psi)$ to transition continuously from 1 to 1/2, filling the gap when $\rho=1/2$.

math.NT↗

arXiv2024

Common substring with shifts in b-ary expansions

Denote by $S_n(x,y)$ the length of the longest common substring of $x$ and $y$ with shifts in their first $n$ digits of $b$-ary expansions. We show that the sets of pairs $(x,y)$, for which the growth rate of $S_n(x,y)$ is $\alpha \log n$ with $0\le \alpha \le \infty$, have full Hausdorff dimension.

math.NT↗

arXiv2024

Metrical theory of power-2-decaying Gauss-like expansion

Each $x\in (0,1]$ can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of \begin{equation*} x=\sum_{i=1}^{\infty}2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))},\qquad d_i(x)\in \mathbb{N}. \end{equation*} Let $\phi:\mathbb{N}\to \mathbb{R}^{+}$ be an arbitrary positive function. We are interested in the size of the set $$F(\phi)=\{x\in (0,1]:d_n(x)\ge \phi(n)~~\text{for infinity many}~n\}.$$ We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of $F(\phi)$. When the Lebesgue measure of $F(\phi)$ is zero, we calculate its Hausdorff dimension. Furthermore, we analyse the growth rate of the maximal digit among the first $n$ digits from probability and multifractal perspectives.

math.NT↗