arXiv subjects

Xianchang Meng

Publications and source records attributed to Xianchang Meng.

14 recordsLinked to original sources

arXiv2023

Visible lattice points in Pólya's walk

In this paper, for any integer $k\geq 2$, we study the distribution of the visible lattice points in certain generalized Pólya's walk on $\mathbb{Z}^k$: perturbed Pólya's walk and twisted Pólya's walk. For the first case, we prove that the density of visible lattice points in a perturbed Pólya's walk is almost surely $1/ζ(k)$, where $ζ(s)$ denotes the Riemann zeta function. A trivial case of our result covers the standard Pólya's walk. Moreover, we do numerical experiments for the second case, we conjecture that the density is also almost surely $1/ζ(k)$.

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arXiv2022

Asymptotic estimate on the distance energy of lattices

Since the well-known breakthrough of L. Guth and N. Katz on the Erdos distinct distances problem in the plane, mainstream of interest is aroused by their method and the Elekes-Sharir framework. In short words, they study the second moment in the framework. One may wonder if higher moments would be more efficient. In this paper, we show that any higher moment fails the expectation. In addition, we show that the second moment gives optimal estimate in higher dimensions.

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arXiv2022

Visible lattice points in higher dimensional random walks and biases among them

For any integers $k\geq 2$, $q\geq 1$ and any finite set $\mathcal{A}=\{{\boldsymbolα}_1,\cdots,{\boldsymbolα}_q\}$, where ${ \boldsymbolα_t}=(α_{t,1},\cdots,α_{t,k})~(1\leq t\leq q)$ with $0<α_{t,1},\cdots,α_{t,k}<1$ and $α_{t,1}+\cdots+α_{t,k}=1$, this paper concerns the visibility of lattice points in the type-$\mathcal{A}$ random walk on the lattice $\mathbb{Z}^k$. We show that the proportion of visible lattice points on a random path of the walk is almost surely $1/ζ(k)$, where $ζ(s)$ is the Riemann zeta-function, and we also consider consecutive visibility of lattice points in the type-$\mathcal{A}$ random walk and give the proportion of the corresponding visible steps. Moreover, we find a new phenomenon that visible steps in both of the above cases are not evenly distributed. Our proof relies on tools from probability theory and analytic number theory.

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arXiv2020

Random walks on generalized visible lattice points

We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker visible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger than a positive constant.

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arXiv2020

Distinct distances on hyperbolic surfaces

For any cofinite Fuchsian group $Γ\subset {\rm PSL}(2, \mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $Γ\backslash\mathbb{H}^2$ determines $\geq C_Γ \frac{N}{\log N}$ distinct distances for some constant $C_Γ>0$ depending only on $Γ$. In particular, for $Γ$ being any finite index subgroup of ${\rm PSL}(2, \mathbb{Z})$ with $μ=[{\rm PSL}(2, \mathbb{Z}): Γ]<\infty$, any set of $N$ points on $Γ\backslash\mathbb{H}^2$ determines $\geq C\frac{N}{μ\log N}$ distinct distances for some absolute constant $C>0$.

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arXiv2020

Erdős distinct distances in hyperbolic surfaces

In this paper, we introduce the notion of "geodesic cover" for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic surface. Then we use estimates of geodesic-covering numbers to study the distinct distances problem in hyperbolic surfaces. Especially, for $Y$ from a large class of hyperbolic surfaces, we establish the nearly optimal bound $\geq c(Y)N/\log N$ for distinct distances determined by any $N$ points in $Y$, where $c(Y)>0$ is some constant depending only on $Y$. In particular, for $Y$ being modular surface or standard regular of genus $g\geq 2$, we evaluate $c(Y)$ explicitly. We also derive new sum-product type estimates.

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arXiv2020

Visible lattice points along curves

This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the "visible" points and the "invisible" points in the definition of previous research. We prove asymptotic formulas for the number of lattice points in different levels of visibility.

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arXiv2020

Chebyshev's bias for products of irreducible polynomials

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.

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arXiv2018

Summatory function of the number of prime factors

We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured that the difference of the summatory functions should attain a constant sign for all sufficiently large $x$. In this paper, we provide strong evidence for Greg Martin's conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.

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arXiv2017

Discrete bilinear Radon transforms along arithmetic functions with many common values

We prove that for a large class of functions $P$ and $Q$, there exists $d\in (0,1)$ such that the discrete bilinear Radon transform $$B^{\rm dis}_{P,Q}(f,g)(n)=\sum_{m\in\mathbb{Z}\setminus\{0\}} f(n-P(m))g(n-Q(m))\frac{1}{m}$$ is bounded from $l^2\times l^2$ into $l^{1+ε}$ for any $ε\in (d,1)$. In particular, the boundedness holds for any $ε\in (0,1)$ when $P$ (or $Q$) is the Euler totient function $ϕ(|m|)$ or the prime counting function $π(|m|)$.

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arXiv2017

Large bias for integers with prime factors in arithmetic progressions

We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A>0$, our result is uniform for $2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $(a_1 \bmod q, \cdots, a_k \bmod q)$, and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.

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arXiv2017

Chebyshev's bias for products of $k$ primes

For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $ω(n)=k$ or $Ω(n)=k$ in two different arithmetic progressions, where $ω(n)$ is the number of distinct prime factors of $n$ and $Ω(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $Ω(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with $ω(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases.

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arXiv2016

Simultaneous distribution of fractional parts of Riemann zeta zeros

We investigate the simultaneous distribution of the fractional parts of $\{α_1 γ, α_2γ, \cdots, α_nγ\}$, where $n\geq 2$, $α_1$, $α_2$, $\ldots$, $α_n$ are fixed, distinct positive real numbers and $γ$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta-function.

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arXiv2016

The distribution of $k$-free numbers and the derivative of the Riemann zeta-function

Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $ζ(s)$. Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}μ_k(n)-\frac{x}{ζ(k)}$ and $μ_k(n)$ is the characteristic function of $k$-free numbers. Finally, we make a conjecture about the maximum order of $M_k(x)$ by heuristic analysis on the tail of the limiting distribution.

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