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Qing-Long Zhou

Publications and source records attributed to Qing-Long Zhou.

9 recordsLinked to original sources
arXiv2026

Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution

This paper investigates inhomogeneous Poissonian pair correlation (PPC), its weak form, and equidistribution. We establish that weak inhomogeneous PPC does not imply inhomogeneous PPC. Furthermore, we construct a sequence satisfying weak inhomogeneous PPC that fails to be equidistributed, which stands in sharp contrast to the homogeneous case. Finally, we prove that for distinct $γ_1, γ_2 \in (0, \frac{1}{2}]$, weak $γ_1$-PPC does not imply weak $γ_2$-PPC, showing that different inhomogeneous parameters give rise to mutually independent notions of weak inhomogeneous PPC.

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arXiv2026

Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture

Let \(Q \subseteq \mathbb{N}\) be a subset, and let \(ψ\colon \mathbb{N} \to [0, \tfrac{1}{2})\), \(θ\colon \mathbb{N} \to \mathbb{R}\) be functions. Let \(\{A_q\}\) and \(\{B_q\}\) be sequences of integers such that \(\gcd(A_q, B_q) = 1\) and \(B_q > 0\) for all \(q\). Define \(W_Q^{\ast}(ψ,θ)\) to be the set of \(x \in [0,1]\) for which \[ \left| x - \frac{p + θ(q)}{q} \right| < \frac{ψ(q)}{q} \] holds for infinitely many \((p,q) \in \mathbb{Z} \times Q\) with \(\gcd(B_q p + A_q, q) = 1\). In this paper, we determine the Fourier dimension of \(W_Q^{\ast}(ψ,θ)\). Our result not only recovers the classical theorems of Kaufman and Bluhm (concerning the homogeneous case \(ψ(q) = q^{-τ}\) with \(τ\ge 1\)) and the one-dimensional version of a result by Cai and Hambrook on the inhomogeneous approximable set, but also provides a complete inhomogeneous generalization. Moreover, it gives an affirmative answer to the coprime formulation of the Chen--Xiong conjecture.

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arXiv2025

Twisted Diophantine approximation for matrix transformations of tori

Consider a sequence of integral matrices $\mathcal{A}=(A_n)_{n\in\N}$, and a $d$-tuple function ${\bf r}=(r_1,\ldots,r_d)\colon \N\to (0,\frac{1}{2})$. For a fixed vector ${\bm α},$ we are interested in the set $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r})$ of vectors ${\bm β}\in[0,1)^{d}$ for which $A_n{\bm α}~~\!\!\!\!\!\pmod{1}$ infinitely often lies in the box centred at ${\bm β}$, with side lengths $2r_i(n)$ in each coordinate direction. Under mild conditions on $\mathcal{A}$ and ${\bf r}$, we prove a metric dichotomy for the size of $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}),$ valid for almost every ${\bm α}$ with respect to any fractal measure with a certain polynomial Fourier decay rate. Furthermore, removing all restrictions on ${\bf r}$, we establish a metric dichotomy for Lebesgue almost every ${\bm α}.$ This solves a variant of a conjecture of González Robert, Hussain, Shulga and Ward [Conjecture 1.10, Bull. London Math. Soc. 2025]. Finally, we also establish a Jarník-type theorem for $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}).$

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arXiv2025

Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets

Let $E\subset [0,1)^{d}$ be a set supporting a probability measure $μ$ with Fourier decay $|\widehatμ({\bf{t}})|\ll (\log |{\bf{t}}|)^{-s}$ for some constant $s>d+1.$ Consider a sequence of expanding integral matrices $\mathcal{A}=(A_n)_{n\in\N}$ such that the minimal singular values of $A_{n+1}A_{n}^{-1}$ are uniformly bounded below by $K>1$. We prove a quantitative Schmidt-type counting theorem under the following constraints: (1) the points of interest are restricted to $E$; (2) the denominators of the ``shifted'' rational approximations are drawn exclusively from $\mathcal{A}$. Our result extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to the matrix setting, advancing the study of Diophantine approximation on fractals. Moreover, it strengthens the equidistribution property of the sequence $(A_n{\bf x})_{n\in\N}$ for $μ$-almost every ${\bf x}\in E.$ Applications include the normality of vectors and shrinking target problems on fractal sets.

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arXiv2024

Non-Salem sets in multiplicative Diophantine approximation

In this paper, we answer a question of Cai-Hambrook in (arXiv$\colon$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $ψ$-well approximable set $$M_2^{\times}(ψ)=\left\{(x_1,x_2)\in [0,1]^{2}\colon \|qx_1\|\|qx_2\|<ψ(q) \text{ for infinitely many } q\in \N\right\},$$ where $ψ\colon\N\to [0,\frac{1}{4})$ is a positive function satisfying $\sum_qψ(q)\log\frac{1}{ψ(q)}<\infty.$ As a corollary, we show that the set $M_2^{\times}(q\mapsto q^{-τ})$ is non-Salem for $τ>1.$

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arXiv2024

Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers

Let $E\subset [0,1]$ be a set that supports a probability measure $μ$ with the property that $|\widehatμ(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension.

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arXiv2023

Metrical properties for the large partial quotients with product forms in continued fractions

The metrical theory of the product of consecutive partial quotients is associated with the uniform Diophantine approximation, specifically to the improvements to Dirichlet's theorem. Achieving some variant forms of metrical theory in continued fractions, we study the distribution of the at least two large partial quotients with product forms among the first $n$ terms. More precisely, let $[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of an irrational number $x\in(0,1),$ and let $φ\colon \N\to\R$ be a non-decreasing function, we completely determine the size of the set \begin{align*} \mathcal{F}_2(φ)=\Big\{x\in[0,1)\colon \exists ~1\le k\neq l \le n, ~&a_{k}(x)a_{k+1}(x)\ge φ(n), \\&a_{l}(x)a_{l+1}(x)\ge φ(n) \text{ for infinitely many } n\in \N \Big\} \end{align*} in terms of Lebesgue measure and Hausdorff dimension.

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arXiv2023

Uniform Diophantine approximation and run-length function in continued fractions

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hatν,$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hatν)=\Big\{x\in[0,1)\colon \forall N\gg1, \exists~ n\in[1,N], \text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\hatν}\Big\}$$ for algebraic irrational points $y\in[0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.

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