arXiv subjects

Guo-Dong Hong

Publications and source records attributed to Guo-Dong Hong.

8 recordsLinked to original sources

arXiv2026

Simultaneous popular polynomial differences over finite fields

Green's popular difference theorem says that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $α$, there exists a nonzero $d\in\mathbb F_p$ such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq α^3-\varepsilon. \] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $α$, there exists a nonzero $d\in\mathbb F_p$ such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{ω_i} \geq α^{1+\sum_iω_i}-\varepsilon \] simultaneously for every $ω=(ω_1,\dots,ω_k)\in\{0,1\}^k$. We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime $p$, there is a constant $c>0$ such that, for all sufficiently large $n$, one can find a set $A\subseteq\mathbb F_p^n$ of density $1/2+o_n(1)$ satisfying \[ \max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c. \] That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both $d$ and $2d$ are simultaneously popular differences for three-term arithmetic progressions is false.

math.NT↗

arXiv2026

On the Peres--Schlag orthogonal projection problem and Kakeya-type sets

We investigate the Peres--Schlag nonempty interior problem for orthogonal projections in both the finite-field and Euclidean settings. Over finite fields $\mathbb F_q^n$, we employ the polynomial method to establish sharp projection results, and uncover a new connection with stability versions of the finite-field $(n,m)$-set problem. Over Euclidean spaces $\mathbb R^n$, we obtain improved nonempty interior results beyond those of Peres and Schlag in certain parameter ranges. Our proof combines techniques from geometric measure theory and harmonic analysis, including $L^p$-estimates for Kakeya maximal operators and maximal $k$-plane transforms.

math.CA↗

arXiv2026

Peres--Schlag's nonempty-interior problem and a shifted-product variant for product sets

We study finite-field analogues of the Peres--Schlag nonempty-interior problem for product sets. Given $A\subseteq\mathbb F_p$, we ask when a suitable one-dimensional linear image of $A^n$ is full; equivalently, when there exist coefficients $t_1,\ldots,t_n\in\mathbb F_p$ such that \[ t_1A+\cdots+t_nA=\mathbb F_p. \] For $n\ge3$, we prove that, for every $η>0$, this holds whenever \[ |A|\gg_{n,η} p^{\frac{3}{2n-1}+η}. \] This improves the exponent predicted by the direct product-set analogue of the Peres--Schlag threshold, namely $|A|\gg p^{2/n}$. We also prove a two-dimensional near-half-density result. Motivated by sum-product phenomena, we also introduce and study a product-type variant in which linear forms are replaced by shifted product maps. We prove finite-field covering results for shifted products \[ (t_1 + A)(t_2 + A)\cdots(t_n + A) \] at the same density scale as in the linear case. Finally, we prove a Euclidean shifted-product analogue: if $A\subseteq\mathbb R$ is Borel and $\dim_H A>2/n$, then some shifted product of $n$ copies of $A$ contains a nonempty open interval.

math.CO↗

arXiv2025

Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus

Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $ε=ε(\mathbb{P})>0$ such that, for any compact set $E \subset \mathbb{T}$ with dim(E)$>1-ε$, we can find $y\neq 0$ so that $\{x,x+P_1(y), \cdots,x+P_k(y)\} \subset E$. The proof relies on a suitable version of the Sobolev smoothing inequality with ideas adapted from Peluse \cite{P19}, Durcik and Roos \cite{DR24}, and Krause, Mirek, Peluse, and Wright \cite{KMPW24}. As a byproduct of our Sobolev smoothing inequality, we demonstrated that the divergence set of the pointwise convergence problem for certain polynomial multiple ergodic averages has Hausdorff dimension strictly less than one.

math.CA↗

arXiv2025

On weighted multilinear polynomial averages in finite fields

We study the weighted multilinear polynomial averages in finite fields. The essential ingredient is the $u^s$-norm control of the corresponding weighted multilinear polynomial averages in finite fields, which is motivated by Teräväinen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of $\mathbb{F}_p^D$: \[ \textbf{x}, \textbf{x}+ P_1(φ(y))v_1,\cdots, \textbf{x}+ P_k(φ(y))v_k, \] where $\mathbb{V}=\{v_1, \cdots, v_{k} \in \mathbb{Z}^D\}$ is a collection of nonzero vectors, $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{Z}[y]\}$ is a collection of linearly independent polynomials with zero constant terms, and $φ(y) \in \mathbb{Q}(y)$ is a nonzero rational function.

math.NT↗

arXiv2024

A study guide for "On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" after T. Orponen and P. Shmerkin

This article is a study guide for ``On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane" by Orponen and Shmerkin. We begin by introducing Furstenberg set problem and exceptional set of projections and provide a summary of the proof with the core ideas.

math.CA↗

arXiv2024

Three term rational function progressions in finite fields

Let $F(t),G(t)\in \mathbb{Q}(t)$ be rational functions such that $F(t),G(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the three term rational function progressions of the form $x,x+F(y),x+G(y)$ in subsets of $\mathbb{F}_p$. The main new ingredient is an algebraic geometry version of PET induction that bypasses Weyl's differencing. This answers a question of Bourgain and Chang.

math.NT↗

arXiv2021

The Group Action Method and Radial Projection

The group action methods have been playing an important role in recent studies about the configuration problems inside a compact set $E$ in Euclidean spaces with given Hausdorff dimension. In this paper, we further explore the group action methods to study the radial projection problems for Salem sets.

math.CA↗