arXiv subjects

Diego Marques

Publications and source records attributed to Diego Marques.

At least 19 recordsLinked to original sources

arXiv2026

Mahler's Problem on Liouville Numbers

A classical theorem of Maillet states that every nonconstant rational function with rational coefficients maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether there exists a transcendental entire function with the same property. We resolve this question in the negative. More generally, we prove that every real-analytic function on an interval with this property is the restriction of a rational function in $\mathbb{R}(x)$. The local result is quantitative: for every nonrational real-analytic function, every nonempty open subinterval of its domain contains a Liouville number whose image has irrationality exponent at most $100$. The proof rests on a two-height counting estimate that treats source and target denominators independently. An adaptive determinant argument, uniform Wronskian sublevel bounds, and Farey separation establish the estimate; a nested-interval construction yields the quantitative result.

math.NT↗

arXiv2026

Smooth Diffeomorphisms and Mahler's Problem on Liouville Numbers

A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville numbers, there exist orientation-preserving $C^\infty$ diffeomorphisms $f:\mathbb{R}\to\mathbb{R}$, arbitrarily close to the identity and transcendental over $\mathbb{R}(x)$, such that for every real number field $K\subset\mathbb{R}$, every $n\geq 1$, and every $m\geq 0$, \[ D^m(f^{\circ n})(K)\subseteq K, \qquad D^m(f^{\circ n})(\mathscr{L})\subseteq\mathscr{L}. \] In fact, the non-analyticity locus may be prescribed as any nonempty compact perfect nowhere-dense set disjoint from the real algebraic and Liouville numbers. The proof combines Maillet's theorem with an arithmetic refinement of Körner's smooth polynomial sewing method and a rational-germ construction.

math.NT↗

arXiv2026

A Neighboring-Denominator Variant of the Erdős--Mahler Conjecture

We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $ξ$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(S)>0$ such that \[ \log q_{n+1}\gg_{ξ,S} q_n^c \] along those indices. Consequently, $ξ$ is a Liouville number. The proof uses the determinant identity for consecutive convergents and a fixed-base consequence of Yu's theorem on $p$-adic logarithmic forms.

math.NT↗

arXiv2026

Almost all binary forms of degree $\geq 3$ fail to represent a fixed integer

We prove that for any fixed integer $ n \geq 3 $ and nonzero integer $ m $, the proportion of integral binary forms of degree $ n $ that represent $ m $ tends to zero as the height tends to infinity. In fact, almost all such forms fail to represent $ m $. Our method uses lattice point counting and geometric methods, including Davenport's lemma and estimates for volumes of hyperplane sections of cubes, together with an analysis of the distribution of rational points on such hyperplanes. The result also holds when restricted to irreducible forms.

math.NT↗

arXiv2026

Height Rigidity for Entire Functions

We prove that algebraic values of bounded degree and polynomially bounded height are sparse on rational translates of the graph of a transcendental entire function. More precisely, for fixed $θ\in\overline{\mathbb{Q}}\cap\mathbb{R}$, $D\geq1$, $t>0$, and for every $\varepsilon>0$, only $O(Q^{\varepsilon})$ rationals $r\in[0,1]$ of height at most $Q$ can satisfy simultaneously $[\mathbb{Q}(f(θ+r)):\mathbb{Q}]\leq D$ and $H(f(θ+r))\ll H(r)^t$. Consequently, if these bounds hold for every rational $r$ of sufficiently large height, then $f\in\overline{\mathbb{Q}}[z]$ and $°f\leq t$. As applications, we obtain rigidity results for entire functions taking rational or bounded-degree algebraic values with polynomially controlled arithmetic height. In particular, this excludes the polynomial-denominator scenario that arises naturally in connection with Mahler's problem on Liouville numbers. The proof combines Pila's bounded-degree counting theorem with standard height estimates and a simple geometric analysis of transcendental entire graphs.

math.NT↗

arXiv2026

Perfect Sets of Liouville Numbers with Controlled Self-Powers

We study the arithmetic behavior of self-powers $x^x$ when $x$ is a Liouville number. Using recent ideas on strengthened Liouville approximation, we develop flexible constructions that illuminate how transcendence, Liouville properties, and "large" topological size interact in this setting. As a concrete outcome, we build a perfect set of Liouville numbers of continuum cardinality whose finite sums, finite products, and self-powers all remain Liouville. These results show that rich algebraic and topological structures persist inside the Liouville universe for the map $x\mapsto x^x$.

math.NT↗

arXiv2026

Algebraic values of transcendental power series with geometric coefficient moduli

Let $λ>1$ be a real algebraic number. We construct continuum many power series $f(z)=\sum_{k\geq0}a_kz^k$ of radius of convergence exactly one such that every nonzero coefficient $a_k$ is algebraic and has modulus $λ^m$ for some $m\geq0$. Moreover, for every integer $s\geq0$, the derivative $f^{(s)}$ takes algebraic values at all algebraic points of the open unit disk and is transcendental over $\mathbb{C}(z)$. The proof combines algebraic polygonal cancellation with a sparse polynomial-block argument. This shows that a multiplicative rank-one restriction on coefficient moduli is compatible with algebraicity of the full analytic jet at every algebraic point once algebraic phases are allowed.

math.NT↗

arXiv2026

Values of algebraic functions at Liouville numbers

In 1953 LeVeque proved the existence of $U_m$-numbers by showing that for some specially defined Liouville number $λ$, the $m$th root $λ^{1/m}$ is in $U_m$. In this article we study the following question: let $u$ be an algebraic function of degree $m$ and $λ$ a Liouville number; under which conditions is $u(λ)$ a $U_m$-number? We consider a more refined notion of $\mathcal{L}$-numbers, and show that, under very general assumptions, an algebraic function of degree $m$ takes $U_m$-values at all $\mathcal{L}$-numbers.

math.NT↗

arXiv2026

Arithmetical structure of sumset intersections

The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$. Then $hA \subseteq \bigcap_{q=1}^{\infty} hA_q$ for all $h \geq 1$. Let $\mathcal{H}(A_q) = \{h \geq 1: hA = \bigcap_{q=1}^{\infty} hA_q\}$. The arithmetical structure of the sets $\mathcal{H}(A_q)$ is unknown. It is proved that for every $h_0 \geq 2$ there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1,\ldots, h_0-1\} \subseteq \mathcal{H}(A_q)$ but $h_0 \notin \mathcal{H}(A_q)$ and also that there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1, h_0 \} \subseteq \mathcal{H}(A_q)$ but $\{2,3, \ldots, h_0-1\} \cap \mathcal{H}(A_q) = \emptyset$.

math.NT↗

arXiv2025

Geometric Progressions meet Zeckendorf Representations

Motivated by Erdős' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and Cantor-like sets in standard bases, we study the corresponding problem in the Zeckendorf numeration system. We prove that, for any fixed finite set of forbidden binary patterns, any integers $u\ge 1$, $q\ge 2$, and any window size $M$, the set of exponents $n$ for which the Zeckendorf expansion of $u q^n$ avoids the forbidden patterns within its $M$ least significant digits is either finite or ultimately periodic.

math.NT↗

arXiv2025

Asymptotic error terms in Bonse-type inequalities

Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1 \cdots p_n > p_{n+1}^{n - π(n)}$ for all $n \geq 2$, where $π(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing the exponent $k(n,x) = n - π(n) + \frac{π(n)}{π(\log n)} - x \cdot π(π(n))$, proving the inequality holds for $x = 2$ and $n \geq 8$. In 2022, Marques and Trojovský extended this to $x = 1.4$ for $n \geq 21$ and conjectured its validity for $x = 0.1$ when $n \geq 24,154,953$. This paper confirms the conjecture by analyzing the error term $E_n(x) = \log(p_1 \cdots p_n) - k(n,x) \log p_{n+1}$. Also, we derive the asymptotic expansion to $E_n(x)$ demonstrating that it is positive for all sufficiently large $n$ when $x > -2$. For each $x > -2$, we identify a minimal integer $Ψ(x)$ such that $E_n(x) > 0$ for all $n \geq Ψ(x)$, precisely determining $Ψ(0.1) = 24,154,953$. Additionally, we establish effective upper bounds for $Ψ(x)$ both unconditionally and under the Riemann Hypothesis, with the conditional bounds showing a significant improvement. Our analysis fully resolves the conjecture and characterizes $Ψ(x)$ as a non-increasing, piecewise constant function, exhibiting discontinuities at a discrete set of threshold points. These results advance the understanding of Bonse-type inequalities and their asymptotic behavior.

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arXiv2025

The generalized Bergshoeff-de Roo identification

There are two main approaches to duality covariant first order higher derivative corrections to the heterotic string, one extending the duality structure and the other deforming the gauge transformations. In this paper we introduce a framework from which both approaches can be derived, proving their equivalence and extending them to higher orders.

hep-th↗

arXiv2025

The generalized Bergshoeff-de Roo identification II

We recently introduced a T-duality covariant mechanism to compute all-order higher-derivative interactions in the heterotic string. Here we extend the formalism to account for a two-parameter family of corrections that also include the bosonic string and HSZ theory. We use our result to compute the full second order Double Field Theory (DFT) for generic values of the parameters, including the generalized Green-Schwarz transformation and its invariant action.

hep-th↗

arXiv2025

Tri-vector symmetry of 11 dimensional supergravity

Kaluza-Klein reductions of 11-dimensional supergravity lead to exceptional global symmetries in lower dimensions. Certain non-geometric elements of these symmetries, parameterized by a tri-vector $γ$, are not inherited from the higher-dimensional local symmetries, but represent instead a symmetry enhancement produced by the isometries of the background. Here, we demonstrate how to realize this enhancement in 11 dimensions, as a symmetry principle with constrained parameters. We show that $γ$ transformations exchange the equations of motion of the metric and the three-form with their Bianchi identities, in a closed form, structuring them into tri-vector multiplets. Implementing this principle as an off-shell symmetry of the theory requires the introduction of a hierarchy of dual fields, including a six-form and a dual graviton in the initial levels.

hep-th↗

arXiv2025

Some results on asymptotic versions of Mahler's problems

In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we demonstrate that certain subsets of algebraic numbers are exceptional sets of some transcendental function $f\in\mathbb{Z}\{z\}$ with almost all bounded coefficients.

math.NT↗

arXiv2024

Algebraic periodic points of transcendental entire functions

We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of $k$-periodic algebraic orbits, for all $k\geq 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.

math.NT↗

arXiv2024

Hidden symmetries from extra dimensions

In Kaluza-Klein compactifications, some symmetries of the higher dimensional theory are preserved in lower dimensions, others are broken, and occasionally, there are symmetry enhancements. The symmetries that are enhanced by toroidal compactifications were recently shown to define a symmetry principle with constrained parameters that fixes the action before dimensional reduction. Here we show the opposite: symmetries of the higher dimensional theory that are broken in the reduction process, can actually be realized after dimensional reduction as a global symmetry principle with constrained parameters that fixes couplings in the lower dimensional theory. We implement this principle in pure gravity, half-maximal supergravity and the circle reduction of 11 dimensional supergravity to Type IIA superstring theory. As a further application, we show that it can be used to constrain the quartic Ramond-Ramond couplings in Type IIA superstring theory from the four-point $ζ(3)\, α'{}^3\, t_8 t_8 R^{(-) 4}$ interactions.

hep-th↗

arXiv2024

On Non Invertible Symmetries in Type IIB Supergravity

In this work, we uncover a collection of non invertible topological operators linked to the 0-, 2-, 4- and 6-form symmetries related to the type IIB superstring effective theory. By pinpointing the $\text{SL}(2,\mathbb{Z})$-covariant conserved currents corresponding to these symmetries, we first derive a set of $\text{SL}(2,\mathbb{Z})$ invariant invertible topological operators that encapsulate the integer BPS charges inherent to the theory. Moving forward, by incorporating fractional charges while maintaining gauge invariance, we introduce the non invertible topological operators for each generalized symmetry, and in particular for the $\text{SL}(2,\mathbb{Z})$ 0-form symmetry. Identifying them as a novel kind of symmetries reminiscent of fractional quantum Hall effect (FQHE)-like non invertible operators, we study their action on charged objects and their associated SymTFTs obtained via half (higher) gauging.

hep-th↗