arXiv subjects

Minh Quy Pham

Publications and source records attributed to Minh Quy Pham.

3 recordsLinked to original sources

arXiv2022

Structural theorems on the distance sets over finite fields

Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $Δ(A)$ contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be $\frac{d}{2}$ in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets, the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances.

math.NT↗

arXiv2021

Bounds for volumes of sub-level sets of polynomials and applications

In this paper, we present some explicit exponents in the estimates for the volumes of sub-level sets of polynomials on bounded sets, and applications to the decay of oscillatory integrals and the convergent of singular integrals.

math.CA↗

arXiv2021

On the $k$-resultant modulus set problem on varieties over finite fields

Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erdős-Falconer distance problem: If $|A|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then we have \[Δ_{k}(A)=\{|x_1+\cdots+x_k|\colon x_i\in A\}\supseteq \mathbb{F}_q^*.\] In this paper, we provide improvements and extensions of their result.

math.NT↗