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Quang-Khai Nguyen

Publications and source records attributed to Quang-Khai Nguyen.

6 recordsLinked to original sources
arXiv2026

Spectrum of the refined Diophantine exponent

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin--Shapiro, and Thue--Morse words, words coming from coding a rotation by intervals, and bracket words.

math.CO↗
arXiv2026

Transcendence and measures via the refined Diophantine exponent

In 2007, Adamczewski and Bugeaud introduced the notion of the Diophantine exponent of an infinite word as a quantitative measure of repetition, leading to new transcendence results for real numbers whose expansions in an integer base are sufficiently simple. In the present article, we introduce the refined Diophantine exponent, which detects weaker forms of repetition while preserving the full strength of the classical approach. This new exponent applies in situations where repetition is partially obscured by some form of noise. Related ideas already appear in the work of Corvaja and Zannier in 2002 and, more recently, in the works of Kebis, Luca, Ouaknine, Scoones, and Worrell. Our approach provides a unified framework that recovers and extends these results, as well as those of Adamczewski and Bugeaud. We also develop quantitative refinements of this method, leading to results about transcendence measures. The recent breakthrough of Bell, Diller, and Jonsson in the context of algebraic dynamics is partly based on a similar idea, which also served as a motivation for the present work.

math.NT↗
arXiv2026

On the generating series of the degree sequence

We study the generating series associated with the degree sequence of a monomial self-map of a projective toric variety. We establish conditions under which this series has its circle of convergence as a natural boundary, and hence is a transcendental, non-holonomic function. In the case of toric surfaces, our results are sharp; moreover, we answer a question of Bell by proving that its reduction modulo $p$ is transcendental for all but finitely many prime numbers $p$.

math.DS↗
arXiv2026

Remarks on a theorem of Silverman

Motivated by a theorem of Silverman, we consider the following problem. Let $A$ be an abelian variety over a global field $K$. Given a non-torsion point $P \in A(K)$, for a sufficiently large positive integer $n$, whether there exists a place $v$ of $K$ such that the order of the reduction of $P$ modulo $v$ is $n$? In this article, we first show that this holds for an elliptic curve over a global function field of positive characteristic $p>3$ and for sufficiently large positive integers $n$ coprime to $p$. In the second part of the paper, we consider its relative version over $\mathbb C$. More precisely, let $π: \mathcal{A} \rightarrow S$ be an abelian scheme over some variety $S$ over $\mathbb C$, and let $P$ be a non-torsion section of $π$. If the Betti map associated to $P$ is generically submersive, then for every sufficiently large $n$, there is a point $s$ in $S(\mathbb C)$ such that $P(s)$ is a point of order $n$ in the corresponding fiber.

math.NT↗
arXiv2026

Abelian varieties are de Rham $K(π,1)$

Motivated by the work of Esnault-Hai, one has the notion of de Rham $K(π,1)$ schemes, defined as follows. Given a smooth proper geometrically connected scheme $X$ over a field $k$ of characteristic 0 and a base point $x \in X (k)$, one can define its differential fundamental group $π^{\mathrm{diff}}(X/k)$, which comes from the Tannakian duality of the category of coherent integrable connections on $X$. Using the formalism of $δ$-functors, one can define natural morphisms between the group-scheme cohomology of $π^{\mathrm{diff}}(X/k)$ and the de Rham cohomology of $X$. One says that $X$ with $x\in X(k)$ is de Rham $K(π,1)$ if such morphisms are all isomorphisms. In this article, we first prove that abelian varieties in characteristic $0$ are de Rham $K(π,1)$. In the second part of the article, we study the group-scheme cohomology of the abelianization of the differential fundamental group of a smooth proper geometrically connected scheme via its Albanese variety.

math.AG↗