Diophantine Approximation Problem Research Articles

JARNÍK TYPE THEOREMS ON MANIFOLDS

AbstractLet $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

MULTIPLICATIVE DIOPHANTINE APPROXIMATION OF THE EXPANSIONS UNDER DIFFERENT BASES ON A LINE

In this paper, we discuss a problem of multiplicative Diophantine approximation of the expansions under different bases restricted on a line. More specifically, let [Formula: see text] be a positive function. A dichotomy law of the Hausdorff measure for the following set: [Formula: see text] is obtained, which depends the convergence or divergence of certain series.

Farey Sequences for Thin Groups

Abstract The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and thin subgroups, they can be used to study interesting number theoretic sequences—for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute and the gap distribution converges and answer an associated problem in Diophantine approximation. Moreover, for one example, we derive an explicit formula for the gap distribution. For this example, we construct the analogue of the Gauss measure, which is ergodic for the Gauss map. This allows us to prove a theorem about the associated Gauss–Kuzmin statistics.

A Practical Diophantine Approximation for Sparse Linear Array Direction Finding

This article presents a computationally efficient, fixed complexity algorithm for estimating the angle of arrival of a radio frequency signal that impinges on a sparse linear array of sensors. The likelihood function for the bearing estimation problem is reduced to an inhomogeneous Diophantine approximation problem involving an unknown set of integer multiples of 2π. In solving for the integers, we propose a modification to the Cassels algorithm and generate an approximate linearized maximum-likelihood (ML) estimate of bearing. The number of iterations for convergence of the algorithm can be precomputed, which allows the algorithm to have fixed computational complexity. The proposed technique applies to a large class of array configurations, allowing the array configuration to be chosen to optimize estimation performance rather than to suit the algorithm. The simulation results show the performance of the proposed bearing estimation algorithm to be very close to that of ML estimation, but with a significantly reduced computational overhead.

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.

Fractal geometry of the complement of Lagrange spectrum in Markov spectrum

The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement M\setminus L of the Lagrange spectrum L in the Markov spectrum M remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that M\setminus L contains infinite countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of M\setminus L were < \sqrt{12} = 3.46\ldots , and this was the status quo of our knowledge of M\setminus L until 2017. In this article, we show the following two results. First, we prove that M\setminus L is richer than it was previously thought because it contains a Cantor set of Hausdorff dimension larger than 1/2 near 3.7: in particular, this solves (negatively) Cusick's conjecture mentioned above. Secondly, we show that M\setminus L is not very thick: its Hausdorff dimension is strictly smaller than one.

Markov random walks on homogeneous spaces and Diophantine approximation on fractals

In the first part, using the recent measure classification results of Eskin–Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space G / Γ G/\Gamma . Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of Simmons–Weiss, we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so, in particular, well approximable.

Bounded and divergent trajectories and expanding curves on homogeneous spaces

Suppose g t g_t is a 1 1 -parameter A d \mathrm {Ad} -diagonalizable subgroup of a Lie group G G and Γ > G \Gamma > G is a lattice. We study the dimension of bounded and divergent orbits of g t g_t emanating from a class of curves lying on leaves of the unstable foliation of g t g_t on the homogeneous space G / Γ G/\Gamma . We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of S L ( 2 , R ) \mathrm {SL}(2,\mathbb {R}) inside G G , which are not contained in a proper parabolic subgroup of G G . We describe applications of our results to problems in Diophantine approximation by number fields and intrinsic Diophantine approximation on spheres. Our methods also yield the following result for lines in the space of square systems of linear forms: suppose φ ( s ) = s Y + Z \varphi (s) = sY + Z where Y ∈ G L ( n , R ) Y\in \mathrm {GL}(n,\mathbb {R}) and Z ∈ M n , n ( R ) Z\in M_{n,n}(\mathbb {R}) . Then, the dimension of the set of points s s such that φ ( s ) \varphi (s) is singular is at most 1 / 2 1/2 while badly approximable points have Hausdorff dimension equal to 1 1 .

Symbolic dynamics in mean dimension theory

AbstractFurstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.Math. Syst. Theory1(1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to$\mathbb{Z}^{2}$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of$\mathbb{Z}^{2}$-subshifts with respect to a subaction of$\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of$\mathbb{Z}^{2}$-subshifts in terms of Kolmogorov–Sinai entropy.

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M.Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove "spiraling" results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh and Tseng as well as Kleinbock, Shi and Weiss.

Optimal density for values of generic polynomial maps

We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $|Q(x)-\\xi|<\\epsilon$ for a generic ternary form $Q$ is $|x|\\ll\\epsilon^\{-1\}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable quadratic surfaces, and the values of the polynomial map defined by the generators of the ring of conjugation-invariant polynomials on $M_3(\\Bbb\{C\})$. These results are instances of a general approach that we develop, which considers a rational affine algebraic subvariety of Euclidean space, invariant and homogeneous under an action of a semisimple Lie group $G$. Given a polynomial map $F$ defined on the Euclidean space which is invariant under a semisimple subgroup $H$ of the acting group $G$, consider the family of its translates $F\\circ g$ by elements of the group. We study the restriction of these polynomial functions to the integer points on the variety confined to a large Euclidean ball. Our main results establish an explicit rate of density for their values, for generic polynomials in the family. This problem has been extensively studied before when the polynomials in question are linear, in the context of classical Diophantine approximation, but very little was known about it for polynomial of higher degree. We formulate a heuristic pigeonhole lower bound for the density and an explicit upper bound for it, formulate a sufficient condition for the coincidence of the lower and upper bounds, and in a number of natural examples establish that they indeed match. Finally, we also establish a rate of density for values of homogeneous polynomials on homogeneous projective varieties.

Equidistribution of Farey sequences on horospheres in covers of and applications

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

PKCHD: Towards a Probabilistic Knapsack Public-Key Cryptosystem with High Density

By introducing an easy knapsack-type problem, a probabilistic knapsack-type public key cryptosystem (PKCHD) is proposed. It uses a Chinese remainder theorem to disguise the easy knapsack sequence. Thence, to recover the trapdoor information, the implicit attacker has to solve at least two hard number-theoretic problems, namely integer factorization and simultaneous Diophantine approximation problems. In PKCHD, the encryption function is nonlinear about the message vector. Under the re-linearization attack model, PKCHD obtains a high density and is secure against the low-density subset sum attacks, and the success probability for an attacker to recover the message vector with a single call to a lattice oracle is negligible. The infeasibilities of other attacks on the proposed PKCHD are also investigated. Meanwhile, it can use the hardest knapsack vector as the public key if its density evaluates the hardness of a knapsack instance. Furthermore, PKCHD only performs quadratic bit operations which confirms the efficiency of encrypting a message and deciphering a given cipher-text.

Remarks on Diophantine approximation in function fields

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

Inhomogeneous diophantine approximation with prime constraints

We study the problem of inhomogeneous diophantine approximation under certain primality restrictions.

A mass transference principle for systems of linear forms and its applications

In this paper we establish a general form of the mass transference principle for systems of linear forms conjectured in 2009. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine–Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine–Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira.

A problem in non-linear Diophantine approximation

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.

A Diophantine approximation problem with two primes and one k-th power of a prime

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a k-th power of a prime which was only proved to hold for 1<k<4/3. We improve the k-range to 1<k≤3 by combining Harman's technique on the minor arc with a suitable estimate for the L4-norm of the relevant exponential sum over primes.

Perfectly ordered quasicrystals and the Littlewood conjecture

Linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals. In a previous paper we presented a characterization of linearly repetitive cut and project sets. In this paper we extend the classical definition of linear repetitivity to try to discover whether or not there is a natural class of cut and project sets which are models for quasicrystals which are better than `perfectly ordered'. In the positive direction, we demonstrate an uncountable collection of such sets (in fact, a collection with large Hausdorff dimension) for every choice of dimension of the physical space. On the other hand, we show that, for many natural versions of the problems under consideration, the existence of these sets turns out to be equivalent to the negation of a well-known open problem in Diophantine approximation, the Littlewood conjecture.

WEAK ADMISSIBILITY, PRIMITIVITY, O‐MINIMALITY, AND DIOPHANTINE APPROXIMATION

We generalise M. M. Skriganov's notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov's celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erd\H{o}s, and others. Finally, we use o-minimality to describe large classes of sets