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Boqing Xue

Publications and source records attributed to Boqing Xue.

18 recordsLinked to original sources
arXiv2026

Sharp refined-direction Kakeya estimates in finite Heisenberg groups

Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $\lambda>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geq\lambda \right\} \right| \lesssim_n q^{2n-1}|E|\lambda^{-2n}. \] The second aim is to determine, for every $1\leq u,v\leq\infty$, the sharp exponent of $q$ in the corresponding $\ell^u\to\ell^v$ estimate. More precisely, we prove that \[ A_n^{\mathrm{rd}}(u,v) = \max\left\{ \frac{2n-1}{v},\ 1-\frac1u,\ \frac{2n}{v}-\frac1u,\ 1+\frac{2n}{v}-\frac{2n+1}{u} \right\}. \] The proof combines the polynomial method with multiplicities and a probabilistic covering argument based on the action of the affine symplectic group.

math.CA↗
arXiv2026

Distribution of simplices in the discrete and continuous settings

In this paper, we study the distribution of simplices in both discrete and continuous settings. Let $q$ be an odd prime power, let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^d$, and let $2\leq k\leq d-1$. We prove that every set $E\subset\mathbb F_q^d$ with \[ |E|\geq C_{d,k}q^{\beta_{d,k}}, \qquad \beta_{d,k}= \begin{cases} \displaystyle \frac{d+k}{2}-\frac{k-1}{k+1}, & d-k\ \text{even},\\[2mm] \displaystyle \frac{d+k-1}{2}, & d-k\ \text{odd}, \end{cases} \] determines a positive proportion of all ordered nondegenerate $k$-simplex congruence classes. This improves the previous exponent due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017), and is sharp when $d-k$ is odd. In the Euclidean setting, we prove that if $E\subset\mathbb R^d$ is compact and $\dim_{\mathrm H}(E)>d-1$, then there exists a Frostman probability measure $\mu$, supported on $E$, and a set of pins of full $\mu$-measure such that the pinned distance configuration measure for labeled $(d-1)$-simplices is absolutely continuous at every such pin. We also show that the same conclusion holds when $E\subset\mathbb R^d$ is a compact Salem set with $\dim_{\mathrm H}(E)>k$.

math.NT↗
arXiv2026

On Erdos-Falconer distance problem in even dimensions

Let $q$ be an odd prime power and $\mathbb{F}_q$ be the finite field of order $q$. We prove an extraction theorem for the Erd\H{o}s-Falconer distance conjecture in even dimensions, showing that the conjecture for all even dimensions reduces to the planar case. As consequences, we obtain improved thresholds on the pinned distance problem and the distribution of triangles, achieving new records of $\frac{d}{2}+\frac{1}{4}$ over prime fields and $\frac{d+1}{2}+\frac{1}{10}$ over arbitrary finite fields, respectively.

math.NT↗
arXiv2026

Alternative Entropy Bounds for Perfect Matchings in Bipartite Graphs

We refine Radhakrishnan's entropy proof of the Br\'egman-Minc bound by introducing a terminal-set framework in which selected vertices are revealed last. This gives new degree-sensitive upper bounds for the number of perfect matchings in bipartite graphs and an explicit formula for single-vertex terminal sets. The bounds recover the standard Br\'egman equality family and improve the estimate for certain nonuniform degree sequences. We also obtain a $C_4$-free refinement complementary to the edge-count bound of Araujo, Balogh and Wang.

math.CO↗
arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq 16/5$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). To prove this result, we develop a new method that combines horizontal slicing, cancellation in an affine fourth-moment expansion, a rich-plane decomposition, and incidence estimates. We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erd\H{o}s--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗
arXiv2026

Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications

Let $q$ be an odd prime power. We study Kakeya maximal operators associated with horizontal lines in the finite Heisenberg groups $\mathbb H_n(\mathbb F_q)$. Our principal object is the refined-direction maximal operator, whose parameter records the projective horizontal direction together with the central homogeneous coordinate determined by horizontality. In rank one, we prove \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(\mathcal D_1)} \lesssim q^{\frac{1}{2}} \|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] where the exponent $\frac{1}{2}$ is sharp. Combining this estimate with endpoint bounds and interpolation, we determine the exact mixed-norm growth exponent: \[ A^{\mathrm{rd}}_1(u,v) = \max\left\{ \frac1v,\, 1-\frac1u,\, \frac2v-\frac1u,\, 1+\frac2v-\frac3u \right\}, \qquad 1\le u,v\le\infty. \] As a consequence, if $E\subset\mathbb H_1(\mathbb F_q)$ meets, in at least $m$ points, a horizontal line in each refined direction from $\Omega\subset D_1$, then \[ |E|\gtrsim \frac{m^2|\Omega|}{q}. \] As a benchmark, we also analyze the coarser operator parameterized only by projective horizontal directions and determine its exact $\ell^u\to\ell^v$ growth exponent in every rank. In rank one, this benchmark is established by a self-contained $TT^*$ argument rather than polynomial vanishing, and the same planar estimate reappears as the zero-central-frequency component of the refined-direction proof. The nonzero central frequencies are controlled by Plancherel, character orthogonality, and a bounded-fiber property of an explicit quadratic map. Thus, the sharp refined-direction estimate is obtained by purely Fourier-analytic methods.

math.CO↗
arXiv2025

Entropy Expansion for General Polynomial Images of Frostman Random Variables

We prove dyadic entropy expansion for the observables $X+Y$ and $f(X,Y)$ under Frostman nonconcentration hypotheses on a prescribed, possibly dependent law. For an integer $n\ge1$, write $H_n(Z)=H(\lfloor 2^nZ\rfloor)$ for the base-two Shannon entropy at resolution $2^{-n}$; thus $n$ indexes the fineness of the dyadic discretization. For every $0 4/3$, we obtain $\max\{H_n(X+Y),H_n(f(X,Y))\}\ge (\frac{s_1+s_2}{2}+\varepsilon)n-O(1)$ for an explicit $\varepsilon>0$. Here the baseline averages the two Frostman exponents. We also classify the exceptional coordinate representations and construct obstructions for algebraic affine directions with Frostman constants uniform in the scale.

math.CA↗
arXiv2025

On a theorem of Mattila in the p-adic setting

Let $A, B$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p^r\mathbb{Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory.

math.NT↗
arXiv2024

On the distance problem over finite p-adic rings

In this paper, we study the distance problem in the setting of finite p-adic rings. In odd dimensions, our results are essentially sharp. In even dimensions, we clarify the conjecture and provide examples to support it. Surprisingly, compared to the finite field case, in this setting, we are able to provide a large family of sets such that the distance conjecture holds. By developing new restriction type estimates associated to circles and orbits, with a group theoretic argument, we will prove the $4/3$-parallel result in the two dimensions. This answers a question raised by Alex Iosevich. In a more general scenario, the existence/distribution of geometric/graph configurations will be also considered in this paper. Our results present improvements and extensions of recent results due to Ben Lichtin (2019, 2023). In comparison with Lichtin's method, our approach is much simpler and flexible, which is also one of the novelties in this paper.

math.CO↗
arXiv2023

New-type Quasirandom Groups and Applications

This paper aims to introduce a more general definition of quasirandom groups and generalize several well-known results in the literature in this new setting. More precisely, let $G$ be a semi-direct product of groups and $X\subseteq G$, we provide conditions such that one can find tuples $(x_0, \ldots, x_k)\in X^{k+1}$ satisfying $x_1x_2\ldots x_k=x_0$ or conditions to guarantee that the product set $XX$ grows exponentially. In a special case of the group of rigid-motions in the plane over an arbitrary finite field, our results offer a reasonably complete description of structures of this group.

math.CO↗
arXiv2020

Asymmetric estimates and the sum-product problems

We show two asymmetric estimates, one on the number of collinear triples and the other on that of solutions to $(a_1+a_2)(a_1^{\prime\prime\prime}+a_2^{\prime\prime\prime})=(a_1^\prime+a_2^\prime)(a_1^{\prime\prime}+a_2^{\prime\prime})$. As applications, we improve results on difference-product/division estimates and on Balog-Wooley decomposition: For any finite subset $A$ of $\mathbb{R}$, \[ \max\{|A-A|,|AA|\} \gtrsim |A|^{1+105/347},\quad \max\{|A-A|,|A/A|\} \gtrsim |A|^{1+15/49}. \] Moreover, there are sets $B,C$ with $A=B\sqcup C$ such that \[ \max\{E^+(B),\, E^\times (C)\} \lesssim |A|^{3-3/11}. \]

math.NT↗
arXiv2019

On a Hilbert Space Reformulation of Riemann Hypothesis

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert subspace $\mathcal{H}_0$ spanned by a certain family of functions coincides with $\mathcal{H}$. A condition that a function does not belong to $\mathcal{H}_0^\bot$ is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid $T_\mathbb{N}=\{T_k:\, k\in \mathbb{N}\}$ of operators is exactly $B(\mathcal{H})$. As a result, Riemann hypothesis is true if and only if $\mathcal{H}_0$ is $T_k^\ast$-invariant for all $k\in \mathbb{N}$.

math.NT↗
arXiv2019

Natural Monoids and Non-commutative Arithmetics

We introduce several classes of monoids satisfying up to five axioms and establish basic theories on their arithmetics. The one satisfying all the axioms is named natural monoid. Two typical examples are 1) the monoid $\mathbb{N}$ of natural numbers in the group of positive rationals and 2) a certain monoid $\mathbb{S}$ in one of Thompson's groups. The latter one is non-abelian, which serves as an important example for non-commutative arithmetics. Defining primes in a non-abelian monoid $S$ is highly non-trivial, which relies on a concept we called `castling'. Three types of castlings are essential to grasp the arithmetics on $S$. Multiplicative and completely multiplicative functions are defined. In particular, Möbius function is multiplicative, and Liouville function on a natural monoid is completely multiplicative. The divisor function has a sub-multiplicative property, which induces a non-trivial quantity $τ_0(u)=\lim\nolimits_{n\rightarrow \infty}(τ(u^n))^{1/n}$ in a non-abelian monoid $S$. Moreover, the quantity $Ç(S)=\sup\nolimits_{1\neq u\in S}τ_0(u)/τ(u)$ describes the complexity for castlings in $S$. We show that $Ç(\mathbb{N})=1/2$ and $Ç(\mathbb{S})=1$. The reduced $C^\ast$-algebra of $S$, on which a particular trace can be defined, is also studied. Furthermore, we prove that a natural monoid having finitely many primes is amenable.

math.NT↗
arXiv2013

On sums of sparse prime subsets

For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $δx (\log x)^{-1}$ with $δ\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\, δx (\log\log x)^{-1}$.

math.NT↗
arXiv2012

On sums of subsets of Chen primes

In this paper we show that if $A$ is a subset of Chen primes with positive relative density $α$, then $A+A$ must have positive upper density at least $cαe^{-c^\prime\log(1/α)^{2/3}(\log\log(1/α))^{1/3}}$ in the natural numbers.

math.NT↗