Search ResultsShowing 1-20 of 189
-
arXiv:2505.03384 (Published 2025-05-06)
Transcendence criteria for multidimensional continued fractions
Comments: 23 pagesCategories: math.NTClassical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization of classical continued fractions, introduced by Jacobi, and there are many well-studied open problems related to them. In this paper, we establish transcendence criteria for multidimensional continued fractions. In particular, we show that some Liouville-type and quasi-periodic multidimensional continued fractions are transcendental. We also obtain an upper bound on the naive height of cubic irrationals arising from periodic multidimensional continued fractions and exploit it to prove the transcendence criteria in the quasi-periodic case.
-
Diophantine approximation of multiple zeta-star values
Comments: Any comments are welcomeThe set of multiple zeta-star values is a countable dense subset of the half line $(1,+\infty)$. In this paper, we establish some classical Diophantine type results for the set of multiple zeta-star values. Firstly, we give a criterion to determine whether a number is a multiple zeta-star value. Secondly, we establish the zero-one law for the set of multiple zeta-star value. Lastly, we propose a conjecture for the set of multiple zeta-star values, which strengthens the original zero-one law.
-
arXiv:2502.09539 (Published 2025-02-13)
Erdős's integer dilation approximation problem and GCD graphs
Comments: 47 pages, 1 figureLet $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{\alpha\in\mathcal{A}\cap[1,x]}\frac{1}{\alpha}>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(\alpha, \beta)\in \mathcal{A}^2$ such that $\alpha\neq \beta$ and $|n\alpha-\beta| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erd\H{o}s from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation.
-
arXiv:2502.08408 (Published 2025-02-12)
Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures
Comments: 17 pages, 2 figuresCategories: math.NTThis paper focuses on the metric properties of L\"uroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarn\'ik--Besicovitch Theorem, and the result of Dodson. A supplementary proof is provided for a measure-theoretic statement originally proposed by Tan--Zhou. The Beresnevich--Velani Mass Transference Principle is applied to extend a dimensional result of Cao--Wu--Zhang. A counterexample is constructed, leading to a revision of a conjecture by Tan--Zhou concerning dimension, along with a partial result.
-
arXiv:2502.00731 (Published 2025-02-02)
Diophantine approximation and the subspace theorem
Comments: Version 1: 49 pages. This article originated from material presented at the workshop "The Subspace Theorem and Its Applications", held at the Chennai Mathematical Institute from December 16 to 28, 2024, where the third author was a speakerCategories: math.NTSubjects: 11J87Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results to higher dimensions, with profound implications to Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth's theorem and Schlickewei's refinement of the subspace theorem, with an emphasis on proofs. The arguments presented are classical and approachable for readers with a background in algebraic number theory, serving as a streamlined, yet condensed reference for these fundamental results.
-
arXiv:2411.12009 (Published 2024-11-18)
More on consecutive multiplicatively dependent triples of integers
Comments: 31 pagesCategories: math.NTIn this paper, we extend recent work of the third author and Ziegler on triples of integers $(a,b,c)$, with the property that each of $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ is multiplicatively dependent, completely classifying such triples in case $a=2$. Our techniques include a variety of elementary arguments together with more involved machinery from Diophantine approximation.
-
Extravagance, irrationality and Diophantine approximation
Comments: 19 pagesFor an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotent function is integrable. We show that with respect to a ``Renyi measure'' for the Gauss map with the log of the partial quotent function non-integrable, almost all numbers are Liouville. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are via the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type theorem for Diophantine approximation with respect to ``weak Renyi measures''.
-
arXiv:2409.12826 (Published 2024-09-19)
Dimension of Diophantine approximation and applications
Comments: 41 pagesIn this paper we construct a new family of sets via Diophantine approximation, in which the classical examples are endpoints. Our first application is on their Hausdorff dimension. We show a recent result of Ren and Wang, known sharp on orthogonal projections in the plane, is also sharp on $A+cB$, $c\in C$, thus completely settle this ABC sum-product problem. Higher dimensional examples are also discussed. In addition to Hausdorff dimension, we also consider Fourier dimension. In particular, now for every $0\leq t\leq s\leq 1$ we have an explicit construction in $\mathbb{R}$ of Hausdorff dimension $s$ and Fourier dimension $t$, together with a measure $\mu$ that captures both dimensions. In the end we provide a perspective of Knapp's example and treat our Diophantine approximation as its analog in $\mathbb{R}$, that naturally leads to the sharpness of Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem. These are alternatives of recent examples due to Fraser-Hambrook-Ryou. In particular, to deal with the non-geometric case we construct measures of "Hausdorff dimension" $a$ and Fourier dimension $b$, even if $a<b$. This clarifies some difference between sets and measures.
-
arXiv:2409.07442 (Published 2024-09-11)
Additive Bases: Change of Domain
Comments: 11 pagesWe consider two questions of Ruzsa on how the minimum size of an additive basis $B$ of a given set $A$ depends on the domain of $B$. To state these questions, for an abelian group $G$ and $A \subseteq D \subseteq G$ we write $\ell_D(A) \colon =\min \{ |B|: B \subseteq D, \ A \subseteq B+B \}$. Ruzsa asked how much larger can $\ell_{\mathbb{Z}}(A)$ be than $\ell_{\mathbb{Q}}(A)$ for $A\subset\mathbb{Z}$, and how much larger can $\ell_{\mathbb{N}}(A)$ be than $\ell_{\mathbb{Z}}(A)$ for $A\subset\mathbb{N}$. For the first question we show that if $\ell_{\mathbb{Q}}(A) = n$ then $\ell_{\mathbb{Z}}(A) \le 2n$, and that this is tight up to an additive error of at most $O(\sqrt{n})$. For the second question, we show that if $\ell_{\mathbb{Z}}(A) = n$ then $\ell_{\mathbb{N}}(A) \le O(n\log n)$, and this is tight up to the constant factor. We also consider these questions for higher order bases. Our proofs use some ideas that are unexpected in this context, including linear algebra and Diophantine approximation.
-
arXiv:2409.02970 (Published 2024-09-04)
Central Limit Theorem for Diophantine approximation on spheres
We prove a Central Limit Theorem and an effective estimate for the counting function of Diophantine approximants on the sphere S$^n$ using homogeneous dynamics on the space of orthogonal lattices.
-
arXiv:2408.01314 (Published 2024-08-02)
Diophantine Approximation with Piatetski-Shapiro Primes
Comments: 12 pagesCategories: math.NTWe prove that for every irrational number $\alpha$, real number $c$ satisfying $1<c<9/8$ and positive real number $\theta$ satisfying $\theta<(9/c-8)/10$, there exist infinitely many primes of the form $p=\left[n^c\right]$ with $n\in \mathbb{N}$ such that $||\alpha p||<p^{-\theta}$.
-
arXiv:2407.11525 (Published 2024-07-16)
On a Theorem of Nathanson on Diophantine Approximation
Categories: math.NTIn 1974, M. B. Nathanson proved that every irrational number $\alpha$ represented by a simple continued fraction with infinitely many elements greater than or equal to $k$ is approximable by an infinite number of rational numbers $p/q$ satisfying $|\alpha-p/q|<1/(\sqrt{k^2+4}q^2)$. In this paper we refine this result.
-
arXiv:2407.11166 (Published 2024-07-15)
On a Theorem of Legendre on Diophantine Approximation
Categories: math.NTLegendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left |\alpha-\frac{p}{q} \right | < \frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this paper we refine these results.
-
arXiv:2406.17544 (Published 2024-06-25)
Diophantine approximation with a quaternary problem
Comments: arXiv admin note: text overlap with arXiv:1703.02381Categories: math.NTLet $1<k<7/6$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^k-\omega|\le (\max_j p_j)^{-\frac{7-6k}{14k}+\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$.
-
arXiv:2403.12341 (Published 2024-03-19)
Diophantine approximation by rational numbers of certain parity types
Comments: 22 pages, 6 figuresCategories: math.NTFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.
-
arXiv:2401.05169 (Published 2024-01-10)
Diophantine Approximation in local function fields via Bruhat-Tit trees
Comments: 19 pages and 6 figures. Comments are welcomeCategories: math.NTWe use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the constants involved and, occasionally, explicit examples of badly approximable quadratic irrationals. Additionally, we can use this method to easily compute the measure of the set of elements that can be written as the limit of a sequence of ``better than expected'' approximants. All these results can be easily obtained via continued fractions when they are available, so that quotient graphs can be seen as a partial replacement of them when this fails to be the case.
-
arXiv:2312.15455 (Published 2023-12-24)
Inhomogeneous Kaufman Measures and Diophantine Approximation
Categories: math.NTWe introduce an inhomogeneous variant of Kaufman's measure, with applications to diophantine approximation. In particular, we make progress towards a problem related to Littlewood's conjecture.
-
arXiv:2312.06110 (Published 2023-12-11)
The exceptional set for Diophantine approximation with mixed powers of prime variables
Categories: math.NTLet lambda_1, \lambda_2, \lambda_3, \lambda_4 be non-zero real numbers, not all negative, with \lambda_1/\lambda_2 irrational and algebraic. Suppose that \mathcal{V} is a well-spaced sequence and \delta >0. In this paper, it is proved that for any \varepsilon >0, the number of v \in \mathcal{V} with v \leqslant N for which |\lambda_1 p_1^2 + \lambda_2 p_2^3+ \lambda_3 p_3^4+ \lambda_4 p_4^5 - v| < v^{-\delta} has no solution in prime variables p_1,p_2,p_3,p_4 does not exceed O\big(N^{\frac{359}{378} + 2\delta +\varepsilon}\big). This result constitutes an improvement upon that of Q. W. Mu and Z. P. Gao [12].
-
arXiv:2312.02628 (Published 2023-12-05)
Diophantine approximation with prime denominator in quadratic number fields under GRH
Comments: 30 pagesCategories: math.NTMatom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke $L$-functions.
-
arXiv:2312.02414 (Published 2023-12-05)
Bounds on the gaps in Kronecker sequences (and a little bit more)
We provide bounds on the sizes of the gaps -- defined broadly -- in the set $\{k_1\vbeta_1 + \ldots + k_n\vbeta_n \mbox{ (mod 1)} : k_i \in \Z \cap (0,Q^\frac{1}{n}]\}$ for generic $\vbeta_1, \ldots, \vbeta_n \in \R^m$ and all sufficiently large $Q$. We also introduce a related problem in Diophantine approximation, which we believe is of independent interest.