Diophantine Approximation Research Articles

Singular matrices and geometry at infinity of products of symmetric spaces

Let $\mathbf{k}$ be a number field and $\mathbf{k}_M$ the Minkowski space associated to $\mathbf{k}$. Dirichlet's theorem in Diophantine approximation is generalized to the case of systems of linear forms with coefficients in $\mathbf{k}_M$. We study the set of singular systems in this setting. We generalize the transference principle, Dani's correspondence and give an estimate of the Hausdorff dimension of the set of singular systems from below.

Diophantische Approximationen

This workshop was focused on a large variety of problems which have seen important progress during the last few years, such as extensions and refinements of Schmidt Subspace Theorem, together with new applications, works on the Zilber-Pink conjecture and on unlikely intersections, geometry of numbers, simultaneous Diophantine approximation, theory of heights, continued fractions, and arithmetic dynamics.

Dimensions of popcorn-like pyramid sets

This article concerns the dimension theory of the graphs of a family of functions which include the well-known ‘popcorn function’ and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimensions. As tools in the proofs, we use the Chung—Erdős inequality from probability theory, higher-dimensional Duffin—Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the Hölder distortion between different graphs.

The shrinking target problem for matrix transformations of tori: Revisiting the standard problem

Let T be a d×d matrix with real coefficients. Then T determines a self-map of the d-dimensional torus Td=Rd/Zd. Let {En}n∈N be a sequence of subsets of Td and let W(T,{En}) be the set of points x∈Td such that Tn(x)∈En for infinitely many n∈N. For a large class of subsets (namely, those satisfying the so called bounded property (B) which includes balls, rectangles, and hyperboloids) we show that the d-dimensional Lebesgue measure of the shrinking target set W(T,{En}) is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function R(x,N):=#{1≤n≤N:Tn(x)∈En}. The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of W(T,{En}). In the case the subsets En are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in [25] for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.

Editorial

Editorial

An Analogue of Mahler’s Transference Theorem for Multiplicative Diophantine Approximation

An Analogue of Mahler’s Transference Theorem for Multiplicative Diophantine Approximation

John William Scott (‘Ian’) Cassels. 11 July 1922—27 July 2015

J. W. S. Cassels was a distinguished number theorist, who was Sadleirian Professor in Cambridge from 1967 to 1984. He worked in many areas of the subject. Perhaps his greatest contribution was concerning Elliptic Curves, much of which is described in a series of eight ground-breaking papers. He also made significant contributions to Diophantine Approximation, Geometry of Numbers and Quadratic Forms, as well as writing influential textbooks on all of these areas. Furthermore, he was an early exponent of the use of computers as an experimental tool in number theory.

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

Reformulating the p-adic Littlewood Conjecture in terms of infinite loops mod pk

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p-adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m∈N such that pℓα is an infinite loop mod pm, for all ℓ∈N∪{0}.

Intrinsic Diophantine approximation for overlapping iterated function systems

In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine’s theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of Qd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}^d$$\\end{document} contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.

Diophantine approximation on conics

Given a conic C \mathcal {C} over Q \mathbb {Q} , it is natural to ask what real points on C \mathcal {C} are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz’s theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha–Kim [Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum, arXiv: 1903.02882, 2021] and Cha–Chapman–Gelb–Weiss [Monatsh. Math. 197 (2022), pp. 1–55] shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general C \mathcal {C} and prove that their behavior is exhausted by the special family of conics C n : X Z = n Y 2 \mathcal {C}_n : XZ = nY^2 , which has symmetry by the modular group Γ 0 ( n ) \Gamma _0(n) and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh [Trans. Amer. Math. Soc. 352 (2000), pp. 4067–4094]. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of Mat 2 × 2 ⁡ ( Z ) \operatorname {Mat}^{2\times 2}(\mathbb {Z}) and classifying invariant lattices in its 2 2 -dimensional representation.

Explicit Salem sets in [formula omitted

We construct the first explicit (i.e., non-random) examples of Salem sets in Rn of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 50 years ago. The construction is based on a form of Diophantine approximation in number fields.

Inhomogeneous Diophantine approximation for generic homogeneous functions

This paper is a sequel to [Monatsh. Math. 194 (2021) 523–554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers [Formula: see text] and [Formula: see text], any [Formula: see text], and any homogeneous function [Formula: see text] that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function [Formula: see text] for a generic element [Formula: see text] in the [Formula: see text]-orbit of [Formula: see text] to be (respectively, not to be) [Formula: see text]-approximable at [Formula: see text]: that is, for there to exist infinitely many (respectively, only finitely many) [Formula: see text] such that [Formula: see text] for each [Formula: see text]. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of [Formula: see text] that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace [Formula: see text] above by any closed subgroup of [Formula: see text] that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper.

Alan Baker. 19 August 1939—4 February 2018

Alan Baker single-handedly transformed several areas of number theory. He achieved a major breakthrough in transcendence and applied it to obtain a new and important large class of transcendental numbers, opening the way to the subsequent discovery of several other such classes. He developed quantitative versions and applied them to the effective solutions of many classical diophantine equations as well as the first effective improvement on the 1844 result by Joseph Liouville (ForMemRS 1850) on diophantine approximation and the resolution of the celebrated Gauss Conjectures of 1801 on class numbers, not only h = 1 but also h = 2, of imaginary quadratic fields. He started the study of extensions to elliptic curves, opening the way to later generalizations to abelian varieties and commutative group varieties and in turn their applications to old and new problems in diophantine geometry. In 1970 he was awarded the Fields Medal at the International Congress in Nice on the basis of his outstanding work on linear forms in logarithms and its consequences. He received many other honours, including the prestigious University of Cambridge Adams Prize, and he was made an honorary fellow of University College London, a foreign fellow of the Indian Academy of Science, a foreign fellow of the National Academy of Sciences India, an honorary member of the Hungarian Academy of Sciences and a fellow of the American Mathematical Society.

Simultaneous p-adic Diophantine approximation

AbstractThe aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

Sums of Fibonacci numbers close to a power of 2

In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in logarithms due to Matveev and Dujella–Pethö version of the Baker–Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.

Upper bounds and spectrum for approximation exponents for subspaces ofℝn

This paper uses W. M. Schmidt’s idea formulated in 1967 to generalise the classical theory of Diophantine approximation to subspaces of ℝ n . Given two subspaces of ℝ n , A and B of respective dimensions d and e, with d+e⩽n, the proximity between A and B is measured by t=min(d,e) canonical angles 0⩽θ 1 ⩽⋯⩽θ t ⩽π/2; we set ψ j (A,B)=sinθ j . If B is a rational subspace, its complexity is measured by its height H(B)=covol(B∩ℤ n ). We denote by μ n (A|e) j the exponent of approximation defined as the upper bound (possibly equal to +∞) of the set of β>0 such that for infinitely many rational subspaces B of dimension e, the inequality ψ j (A,B)⩽H(B) -β holds. We are interested in the minimal value μ ˚ n (d|e) j taken by μ n (A|e) j when A ranges through the set of subspaces of dimension d of ℝ n such that for all rational subspaces B of dimension e one has dim(A∩B)<j. We show that if A is included in a rational subspace F of dimension k, its exponent in ℝ n is the same as its exponent in ℝ k via a rational isomorphism F→ℝ k . This allows us to deduce new upper bounds for μ ˚ n (d|e) j . We also study the values taken by μ n (A|e) e when A is a subspace of ℝ n satisfying dim(A∩B)<e for all rational subspaces B of dimension e.

Public key exponent attacks on multi-prime power modulus using continued fraction expansion method

This paper proposes three public key exponent attacks of breaking the security of the prime power modulus 𝑁=𝑝2𝑞2 where 𝑝 and 𝑞 are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus 𝑁=𝑝2𝑞2 for q&lt;𝑝&lt;2q using key equation 𝑒𝑑−𝑘𝜙(𝑁)=1 where 𝜙(𝑁)= 𝑝2𝑞2(𝑝−1)(𝑞−1) can be broken by recovering the secret keys 𝑘 /𝑑 from the convergents of the continued fraction expansion of e/𝑁−2𝑁3/4 +𝑁1/2 . The paper also reports the second and third approaches of factoring 𝑛 multi-prime power moduli 𝑁𝑖=𝑝𝑖2 𝑞𝑖2 simultaneously through exploiting generalized system of equations 𝑒𝑖𝑑−𝑘𝑖𝜑(𝑁𝑖)=1 and 𝑒𝑖𝑑𝑖−𝑘𝜑(𝑁𝑖)=1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for 𝑖=1,2,…,𝑛.

On the metric theory of multiplicative Diophantine approximation

In 1962, Gallagher proved a higher-dimensional version of Khintchine’s theorem on Diophantine approximation. Gallagher’s theorem states that for any non-increasing approximation function ψ: ℕ → (0, 1/2) with $$\sum\nolimits_{q = 1}^\infty {\psi \left( q \right)} $$ and γ = γ′ = 0 the following set $$\left\{ {\left( {x,y} \right) \in {{\left[ {0,\,1} \right]}^2}:\left\| {qx - \gamma } \right\|\left\| {qy - {\gamma ^\prime }} \right\| < \psi \left( q \right)\,\,{\rm{infinitely}}\,{\rm{often}}} \right\}$$ has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on γ, γ′) of the above result. In this paper, we prove an Erdős—Vaaler type result for fibred multiplicative Diophantine approximation. Along the way, via a different method, we prove a slightly weaker version of Chow—Technau’s theorem with the condition that at least one of γ, γ′ is not Liouville. We also extend Chow—Technau’s result for fibred inhomogeneous Gallagher’s theorem for Liouville fibres.

Diophantine approximation with constraints

Following Schmidt, Thurnheer and Bugeaud–Kristensen, we study how Dirichlet’s theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non