arXiv subjects
Publications and source records attributed to Yu-Chen Sun.
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, M\"obius inversion, and the discrete Fourier transform.
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 $R_m\leqslant 192$.
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 $\eta>0$, for all sufficiently large $x$, \[ N(x)\gg_\eta x\exp\left(-(4+\eta)\frac{\log\log\log x}{\log\log x}\log x\right). \]
Let $\Lambda$ 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) \Lambda(|n|) \frac{e^{2\pi i \lambda n}}{n} : \lambda \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 \[ \sup_{\lambda\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)\Lambda(|n|)\frac{e^{2\pi i\lambda n}}{n}\Big| \] 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.
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 $P_{18}$.
We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_{\gamma} := \{p \text{ is prime and } p = \lfloor n^{1/\gamma} \rfloor \text{ for some } n \in \mathbb{N}\} \] and arbitrary nilsequences when $\gamma \in (0,1)$ 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 $\mathcal{P}_{\gamma}$ to any "finite-complexity" system of linear equations, including for the number of $k$-term arithmetic progressions in $\mathcal{P}_{\gamma}$ 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 \[ 1 - 2^{-Ck} < \gamma < 1, \] then the Piatetski-Shapiro primes $\mathcal{P}_{\gamma}$ contain infinitely many non-trivial $k$-term arithmetic progressions. This significantly improves upon the previous range of $\gamma$ obtained by Li and Pan, which is of triple exponential type.
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.
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.
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, $(X, \nu),$ 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 $\nu(X_f) = 1,$ so that for all $\omega \in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 \pi i p_n \theta} f(T^{p_n} \omega) \] converges for all $\theta \in [0,1]$; 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.
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.
In this note, we discuss two problems of S\'{a}rk\"{o}zy (2001). In particular, we prove an optimal result on sumsets of sparse subset of primes.
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 $\mathbb{E} X_j = n^{-\alpha}$, and we restrict to $1 < c < 7/6, \ 0 < \alpha < 1/2$. Then (almost surely) for any measure-preserving system, $(X,\mu,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $\mu$-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.
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 $\Phi(x)$ be a positive function that tends to infinity arbitrarily slowly as $x\to \infty$. We then show that whenever $H_1\geq(\log x)^{\Phi(x)}$ 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 $1\leq h\leq H_2$.
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 $x\in[X,2X]$, \[ \frac1h\sum_{x<n\leq x+h}d_k(n) = \left(\Phi_{\rm G}\left(\frac{A(X)}{\sqrt{k}\log k}\right)+o(1)\right) M_k(x), \] where $\Phi_{\rm G}$ 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 $x$.
Let $\mathcal{S}=\{1^2,2^2,3^2,...\}$ be the set of squares and $\mathcal{W}=\{w_n\}_{n=1}^{\infty} \subset \mathbb{N}$ be an additive complement of $\mathcal{S}$ so that $\mathcal{S} + \mathcal{W} \supset \{n \in \mathbb{N}: n \geq N_0\}$ for some $N_0$. Let $\mathcal{R}_{\mathcal{S},\mathcal{W}}(n) = \#\{(s,w):n=s+w, s\in \mathcal{S}, w\in \mathcal{W}\} $. In 2017, Chen-Fang \cite{C-F} studied the lower bound of $\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)$. In this note, we improve Cheng-Fang's result and get that $$\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/2}.$$ As an application, we make some progress on a problem of Ben Green problem by showing that $$\limsup_{n\rightarrow\infty}\frac{\frac{\pi^2}{16}n^2-w_n}{n}\ge \frac{\pi}{4}+\frac{0.193\pi^2}{8}.$$
In this paper we give a short interval version of the Balog-Ruzsa theorem concerning bounds for the $L_1$ norm of the exponential sum over $r$-free numbers. As an application, we give a lower bound for the $L_1$ norm of the exponential sum defined with the M\"obius function. Namely we show that $$\int_{{\mathbb T}} \left|\sum_{|n-N|<H} \mu(n)e(n \alpha)\right| d \alpha \gg H^{\frac{1}{6}}$$ when $H \gg N^{\frac{9}{17} + \varepsilon}$.
The COVID-19 pandemic is one of the most challenging healthcare crises during the 21st century. As the virus continues to spread on a global scale, the majority of efforts have been on the development of vaccines and the mass immunization of the public. While the daily case numbers were following a decreasing trend, the emergent of new virus mutations and variants still pose a significant threat. As economies start recovering and societies start opening up with people going back into office buildings, schools, and malls, we still need to have the ability to detect and minimize the spread of COVID-19. Individuals with COVID-19 may show multiple symptoms such as cough, fever, and shortness of breath. Many of the existing detection techniques focus on symptoms having the same equal importance. However, it has been shown that some symptoms are more prevalent than others. In this paper, we present a multimodal method to predict COVID-19 by incorporating existing deep learning classifiers using convolutional neural networks and our novel probability-based weighting function that considers the prevalence of each symptom. The experiments were performed on an existing dataset with respect to the three considered modes of coughs, fever, and shortness of breath. The results show considerable improvements in the detection of COVID-19 using our weighting function when compared to an equal weighting function.
We prove that, for any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes $N = p_1 + p_2 +p_3$ such that $p_i = \lfloor n_{i}^{c_i}\rfloor$ for some $n_i \in{\mathbb N}$ for each $1 \leq i \leq 3$. Our arguments are based on a variant of Green's transference principle due to Matom\"aki, Maynard and Shao. We prove a necessary restriction estimate using Bourgain's strategy and employ Harman's sieve method to optimize our upper bound for $c_i$.