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Yu-Chen Sun

Publications and source records attributed to Yu-Chen Sun.

At least 19 recordsLinked to original sources
arXiv2026

On the lower bound for the $L^1$ norm of Liouville exponential sums. I

Let $λ$ be the Liouville function. We prove Let $λ$ be the Liouville function. We prove \[ \mathcal{I}_λ(X)=\int_{\mathbb{T}}\left|\sum_{1\leqslant n\leqslant X}λ\left(n\right)\mathrm{e}\left(nα\right)\right| \mathrm{d}α \gg X^{\frac{1}{4}}. \] This removes the $X^\varepsilon$ loss in the previous result of Pandey and Radziwiłł~\cite{PandeyRadziwill2026}. In the second paper in this series, we will prove that \[ \limsup_{X \to \infty} \frac{\log \mathcal{I}_λ(X)} {\log X} \geqslant\frac{1}{3}. \]

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arXiv2026

VC-1K: A Diverse Video Test Set for Real-Time Communication

We introduce Video Conferencing 1000 (VC-1K), an open-source test set comprising approximately 1,000 video sequences specifically designed for evaluating video processing and encoding pipelines in real-time communication (RTC) scenarios. Unlike existing test sets that primarily rely on studio-produced or screen content, VC-1K consists of user-generated video call content captured on consumer mobile devices under a wide variety of conditions. The dataset predominantly features talking-head sequences with distinct faces, but also includes a small subset of outward-facing, rear-camera videos without faces to reflect additional common RTC use cases. It spans a wide range of real-world variables including device type, lighting conditions, motion levels, indoor/outdoor environments, and demographic diversity. All video sequences are accompanied by synchronous audio to facilitate multimodal research. We detail the dataset's design methodology and statistical distribution, and demonstrate its utility through a comparative encoder evaluation of OpenH264, SVT-AV1, and libaom under constant bitrate (CBR) encoding at resolutions typical of RTC deployments. Furthermore, two super-resolution approaches, RealESRGAN and GFPGAN, are evaluated on VC-1K, highlighting the dataset as a useful benchmark for pre- and post-processing research. VC-1K is intended to serve as a publicly available dataset for the video RTC research community.

eess.IV↗
arXiv2026

Linear equations in Piatetski-Shapiro primes

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_γ := \{p \text{ is prime and } p = \lfloor n^{1/γ} \rfloor \text{ for some } n \in \mathbb{N}\} \] and arbitrary nilsequences when $γ\in (0,1)$ is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in $\mathcal{P}_γ$ to any ``finite-complexity" system of linear equations, including for the number of $k$-term arithmetic progressions in $\mathcal{P}_γ$ up to a threshold $N$ for any given $k \geq 3$. Furthermore, we show that there exists an absolute constant $C>0$ such that if \[ 1 - 2^{-Ck} < γ< 1, \] then the Piatetski-Shapiro primes $\mathcal{P}_γ$ contain infinitely many non-trivial $k$-term arithmetic progressions. This significantly improves upon the previous range of $γ$ obtained by Li and Pan, which is of triple exponential type.

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arXiv2026

Higher additive energies on discrete cubes

Let $m,n\geq2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots,n-1\}^d$ in any dimension $d$ satisfies $E_m(A)\leq |A|^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$ t_{m,n}=2m-1-(1+o(1))\log_n \frac{(2m-1)^{m-1/2}}{(2m)^{m-1}} \qquad(n\to\infty). $$ For $m=2$, this gives $$ t_{2,n} = 3 - (1+o(1)) \log_n\frac{3\sqrt{3}}{4}, $$ proving a conjecture of the first author.

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arXiv2026

The optimal scale in Linnik's microsquare problem for almost all integers

We prove that, for any positive function $Ψ(t)\to\infty$, almost all positive integers $n$ satisfying $4\nmid n$ and $n\not\equiv7\pmod8$ can be written as $n=x^2+y^2+z^2$, where $x,y,z\in\mathbb{Z}$ and $1\leqslant z<\sqrt{\log n}\,Ψ(n)$. This improves Wooley's bound, and we show that the result is best possible. Our approach combines Hooley's weighted representation function for sums of two squares with Green's $W$-trick. We also give a necessary and sufficient condition, in terms of covering systems, for almost all odd integers to be represented as the sum of a squarefree number and a power of two, answering a question of Granville and Soundararajan.

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arXiv2026

Large sieve inequality for sums of Legendre symbols over short intervals

Using the Burgess bound and the Selberg sieve, we obtain an upper bound for the second moment of sums of Legendre symbols over intervals , with the modulus ranging over primes . The bound is nontrivial and yields a power saving in , uniformly for , provided that , where as . This may be viewed as a short-interval analogue of a result of D. R. Heath-Brown (1995) on moments of quadratic character sums over the initial interval . In particular, it implies that, for any prescribed interval of this length, the quadratic residues and non-residues are asymptotically equidistributed for almost all primes . We also establish estimates for higher moments conditionally on the Generalised Riemann Hypothesis. These bounds rely on a sharp uniform estimate for the number of tuples of integers in a shifted interval whose product is a square.

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arXiv2026

The critical-window profile for $d_k$ in short intervals

We establish an almost-all transition theorem for the $k$-fold divisor function in short intervals. Let $k\geq2$ be fixed, let $\ell=\log\log X$, and set \[ D_k(X)=(\log X)^{k\log k-k+1}, \qquad M_k(x)=\frac1x\sum_{x<n\leq2x}d_k(n). \] The critical scale is $D_k(X)$. In the bounded part of the transition window, put \[ A(X)=\frac1{\sqrt{\ell}}\log\frac{h}{D_k(X)} \] and assume that $A(X)=O(1)$. Then, for almost all integers $x\in[X,2X]$, \[ \frac1h\sum_{x<n\leq x+h}d_k(n) = \left(Φ_{\rm G}\left(\frac{A(X)}{\sqrt{k}\log k}\right)+o(1)\right) M_k(x), \] where $Φ_{\rm G}$ denotes the standard Gaussian distribution function. Above the window, namely when \[ \frac{\log(h/D_k(X))}{\sqrt{\ell}}\to+\infty, \] the full long average is recovered for almost all $x$.

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arXiv2026

On the problem of large gcd for disjoint residue classes

Consider $k$ pairwise disjoint residue classes $a_i \pmod{m_i}$. We prove that \[ \max_{1\leq i<j\leq k}\gcd(m_i,m_j) \gg k\exp\!\left(-(2+o(1)) \sqrt{\frac{\log k}{\log\log k}}\right). \] The proof uses a complete graph whose edges are colored by the gcds of the corresponding moduli, together with a structural lemma, a sieve-theoretic partition, Möbius inversion, and the discrete Fourier transform.

math.CO↗
arXiv2026

Cross representations of additive complements of $r$-th powers

Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}_r\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{3/4-o(1)}, $$ which improves the previous bound $N^{1/2}$ due to Ding, Sun, Wang and Xia.

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arXiv2026

The Wiener Wintner Theorem Along the Primes

We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, ν),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $ν(X_f) = 1,$ so that for all $ω\in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 πi p_n θ} f(T^{p_n} ω) \] converges for all $θ\in [0,1]$; above, $\{2 = p_1 < p_2 < \dots\}$ are an enumeration of the primes. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with U^3 theory playing an important role; our $U^3$-estimates for Heath-Brown models of the von Mangoldt function may be of independent interest.

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arXiv2026

An improved upper bound on the Ruzsa number

Let $R_m$ be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_{m}$ with $A+A=\mathbb{Z}_m$ for which the number of ordered solutions of $n=x+y$ with $x,y\in A$ is at most $r$ for every $n\in \mathbb{Z}_m$. In this note we prove that $R_m\leqslant 128$ for every positive integer $m$, improving the previous bound $R_m\leqslant 192$.

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arXiv2026

An improved lower bound for odd integers not of the form $p+2^a+2^b$

Let $x$ be sufficiently large and \[ N(x)=\big|\bigl\{n\le x:n\ \text{is odd and }n\ne p+2^a+2^b \textrm{ with } p \text{ a prime and } a,b\in \mathbb{N}\bigr\}\big|. \] Motivated by Crocker's result \[ N(x)\gg \log\log x, \] Erd\H os repeatedly asked whether there is an absolute constant $c_0$ such that $N(x)>c_0x$. Pan \cite{Pan} proved in 2011 that \[ N(x)\gg x\exp\!\left( -C_0\frac{\log\log\log\log x}{\log\log\log x}\log x \right), \] where $C_0>0$ is an absolute constant. We improve on Pan's result by showing that, given any $η>0$, for all sufficiently large $x$, \[ N(x)\gg_ηx\exp\left(-(4+η)\frac{\log\log\log x}{\log\log x}\log x\right). \]

math.NT↗
arXiv2026

A Variation Norm Carleson Theorem Along the Primes

Let $Λ$ denote the von Mangoldt function; we prove that for each $r > 2$, there exist constants \[ r' < \mathbf{c}(r) < 2 < \mathbf{C}(r), \qquad \lim_{r \to \infty} \mathbf{c}(r) = 1, \ \lim_{r \to \infty} \mathbf{C}(r) = \infty \] so that the discrete variational Carleson operator along the primes \begin{align} \mathcal{V}^r \Big( \sum_{n \neq 0} f(x-n) Λ(|n|) \frac{e^{2πi λn}}{n} : λ\in \mathbb{T} \Big) \end{align} is bounded on $\ell^p$ for all $\mathbf{c}(r) < p < \mathbf{C}(r)$, while the variation is unbounded when $p \leq r'$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator \[ \sup_{λ\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)Λ(|n|)\frac{e^{2πiλn}}{n}\Big| \] is bounded on \(\ell^p(\mathbb Z)\) for the full expected range \(1<p<\infty\). The proof gives a new mechanism for treating modulation-invariant singular integrals after arithmetic sparsification. It combines higher-order Fourier uniformity, a variable-coefficient multi-frequency principle in the spirit of Bourgain, and an additive-combinatorial inverse argument. A key step is a reduction to finite periodic models, where the Ramanujan structure of the major arcs is converted into a sharp estimate for structured atoms by elementary number theory.

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arXiv2026

Representations of positive integers by three almost-prime squares

Let $P_r$ denote an integer with at most $r$ prime factors, counted with multiplicity. It is known that every sufficiently large integer $N$ satisfying $N \equiv 3 \pmod{24}$ and $5 \nmid N$, can be written in the form $N= x_1^2+x_2^2+x_3^2$ where $x_1,x_2,x_3$ are integers. In this paper, we prove that the above representation in the following two different forms (i) $x_1x_2x_3$ is a $P_{67}$-number; (ii) each $x_i$ is a $P_{27}$-number. This result improves on the previous result of Waibel\cite{Wa}, in which $P_{72}$ was obtained in place of $P_{67}$. The proofs combine the higher-dimensional sieve, a Richert-type weighted sieve method introduced by Cai \cite{Cai} with a Bombieri-Vinogradov type result given by Waibel\cite{Wa}. Applying the same method in a one dimensional sieve setting, we also show that every sufficiently large $N$ not of the form $4^k(8l+7)$ can be written in the form \[ N = x^{2} + y^{2} + (2^{a} z)^{2}, \] where $x,y,a,z$ are non-negative integers and $z$ is a $P_{18}$-number. This improves upon a result of Banerjee \cite{Ban} who obtained $P_{118}$ in place of $P_{18}$.

math.NT↗
arXiv2026

Small values of signed harmonic sums and logarithmic means of multiplicative functions

We construct sequences $\{a_n\}_{n\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset\mathbb{N}.$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ with unusually small logarithmic partial sums, that is, \[ \sum_{n \leq N}\frac{f(n)}{n} \ll \exp\left(-c_0 \frac{N^{1/3}}{(\log N)^{1/3}} \right) \] holds for infinitely many $N\to\infty$. The proofs combine careful analysis of the small-scale distribution of random harmonic sums over subsets of $\mathbb{N}$, together with deterministic inductive arguments inspired by the ``anatomy" of integers.

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arXiv2026

Quantitative Convergence for Sparse Ergodic Averages in $L^1$

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E} X_j = n^{-α}$, and we restrict to $1 < c < 7/6, \ 0 < α< 1/2$. Then (almost surely) for any measure-preserving system, $(X,μ,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $μ$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c = \frac{1001}{1000}, \ \frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting.

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arXiv2025

Local divisor correlations in almost all short intervals

Let $ k,l \geq 2$ be natural numbers, and let $d_k,d_l$ denote the $k$-fold and $l$-fold divisor functions, respectively. We analyse the asymptotic behavior of the sum $\sum_{x 0$ be a small fixed number and let $Φ(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{Φ(x)}$ and $(\log x)^{1000k\log k}\leq H_2\leq H_1^{1-\varepsilon }$, the expected asymptotic formula holds for almost all $x\in[X,2X]$ and almost all $1\leq h\leq H_2$.

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