arXiv subjects

Minh-Quy Pham

Publications and source records attributed to Minh-Quy Pham.

4 recordsLinked to original sources

arXiv2026

On Hausdorff dimensions of $k$-point configuration sets and Elekes-Rónyai type theorems

We prove a ''dimension expansion'' version of the Elekes-Rónyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is true: whenever a Borel set $A\subset\mathbb{R}$ has Hausdorff dimension $α\in \left(\frac{1}{2},1\right)$, $f(A\times A\times A)$ has dimension significantly larger than that of $A$, i.e. \begin{align*} \dim_Hf(A\times A\times A)\geq α+\varepsilon(α),\quad \text{for some } \varepsilon(α)>0, \end{align*} Moreover, if $α>\frac{2}{3}$, $f(A\times A\times A)$ has positive Lebesgue measure. This is a considerable extension of the result established by Koh, T. Pham, and Shen (J. Funct. Anal. 286 (2024)). We also obtain an alternative proof and an improvement for the Elekes-Rónyai type theorem for bivariate real analytic functions established by Raz and Zahl (Geom. Funct. Anal. 34 (2024)). We derive these from more general results, showing that various $k$-point configuration sets of thin sets have positive Lebesgue measure by exploiting the optimal $L^2$-based Sobolev estimates for the associated family of Fourier integral operators. Extending the framework developed by Greenleaf, Iosevich, and Taylor (Mathematika 68 (2022), Math. Z. 306 (2024)) to prove Mattila-Sjölin type theorems, we obtain Falconer-type results for many configuration sets on which the method would be vacuous if demanding nonempty interior. In particular, when $k=2$, we generalize the Falconer-type result for metric functions in $\mathbb{R}^d$ satisfying strong non-vanishing curvature conditions established by Eswarathasan, Iosevich, and Taylor (Adv. Math. 228 (2011)) and the asymmetric Mattila-Sjölin type results of Greenleaf, Iosevich, and Taylor (J. Geom. Anal. 31 (2021)) to a broader class of smooth functions of asymmetric form.

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arXiv2026

On Falconer type functions and the distance set problem

We study the distance set problem for pairs of compact sets $A, B\subset \mathbb{R}^n$, $n\geq 2$. We show that if $B$ is contained in a hyperplane and \begin{align*} \dim_{H} A+\dim_{H} B>n, \end{align*} then the distance set $ Δ(A,B):=\left\{ \vert x-y\vert: x\in A, y\in B\right\}$ has positive Lebesgue measure, and the dimensional threshold is sharp. This yields new positive results for Falconer's distance problem in certain regimes, particularly where the best known bounds fail to apply. We further establish Falconer's distance conjecture for certain classes of product sets under additional structural assumptions. Specifically, if $A=A_1\times A_2\subset \mathbb{R}^{m}\times \mathbb{R}^{n-m}$ for some $0\leq m\leq n-1$, where $A_2$ is a Salem set, and \[ \dim_HA>\frac{n}{2}, \] then the distance set $Δ(A):=\left\{|x-y|: x,y\in A\right\} $ has positive Lebesgue measure. A key feature of our argument is the interpretation of the original map as a suitable projection. We extend the analysis to a broad class of smooth functions, recovering the sharp result of Koh, Pham, and Shen (J. Funct. Anal. 286 (2024)) for quadratic polynomials in three variables.

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arXiv2025

$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve

For $0 < α\leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-α}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number of $\varepsilon$-balls needed to cover $E$. We proved that if $f \in L^p(\mathbb{R}^d)$ with \begin{align*} 1 \leq p\leq p_α:= \begin{cases} \frac{d^2+d+2α}{2α} & d \geq 3, \frac{4}α &d =2, \end{cases} \end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem.

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arXiv2024

Packing sets in Euclidean space by affine transformations

For Borel subsets $Θ\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*} Θ(E):=\bigcup_{(g,z)\in Θ}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and Hausdorff dimension of $Θ(E)$ given the dimensions of the Borel sets $E$ and $Θ$, when $Θ$ has product form. We also study this question by replacing rigid motions with the class of dilations and translations; and similarity transformations. The dimensional thresholds are sharp. Our results are variants of some previously known results in the literature when $E$ is restricted to smooth objects such as spheres, $k$-planes, and surfaces.

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