Number Theory Meets Wireless Communications: An Introduction for Dummies Like Us

Beresnevich, Victor; Velani, Sanju

doi:10.1007/978-3-030-61303-7_1

Cited by 5 publications

(5 citation statements)

References 52 publications

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“…Our new Diophantine appplication is motivated by [BV,§2.7], where Beresnevich and Velani introduced the notion of jointly singular k-tuples of matrices. Namely, say that (Y 1 , .…”

Section: Now We Inductively Prove the Followingmentioning

confidence: 99%

Dimension drop for diagonalizable flows on homogeneous spaces

Kleinbock¹,

Mirzadeh²

2022

Preprint

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Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, let O be an open subset of X, and let F = {gt : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F -orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Ad-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.

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“…Our new Diophantine appplication is motivated by [BV,§2.7], where Beresnevich and Velani introduced the notion of jointly singular k-tuples of matrices. Namely, say that (Y 1 , .…”

Section: Now We Inductively Prove the Followingmentioning

confidence: 99%

Dimension drop for diagonalizable flows on homogeneous spaces

Kleinbock¹,

Mirzadeh²

2022

Preprint

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“…After renormalization, we have Π g = g and µ Π = µ; in particular, µ is supported on Λ S . (B3) Spectral gap 6 : We have e −nP L n − cgµ 1 ≤ Cγ n for some C < ∞ and γ < 1, where c = 1/µg, and • 1 indicates that the operator norm is being taken with respect to the space B = Lip(V C ), rather than the space C(V C ) that L was originally defined on. (B4) Lyapunov exponent: Using the formulas L Π = Π L, Π g = g, and µ Π = µ, we get…”

Section: Symbolic and Geometric Perron-frobenius Operatorsmentioning

confidence: 99%

“…Their influential investigations have led to a blossoming area broadly known as metric Diophantine approximation, with several connections to classical number theoretic questions, as well as more surprising links to mathematical physics, dynamical systems, fractal geometry, analytic combinatorics, computer science, wireless communication, etc. -see [6,7,13,23,27,29,33,44,48,57,62,64,71,81,84] and the references therein for a sampling of such relationships.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

Das¹,

Fishman²,

Simmons³

et al. 2020

Preprint

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We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai-Ruelle-Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all ≤ N ; as well as its counterpart for reals whose partial quotients are all ≥ N due to Good from 1941. CONTENTS1. Introduction 2 2. An abstract operator formula 5 3. Point-accumulating conformal iterated function systems 8 4. Statement of main result 14 5. Spectral gap in the case c > 0 17 6. Proof of Theorem 4.1 19 7. Simplifications in special cases 24 8. Solving (4.1): a motivating computation 28 9. Gauss IFS Examples 30 10. Directions to further research 41 Appendix A. Computation of coefficients 42 Appendix B. Definition of a conformal iterated function system (CIFS) 45 References 46

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“…In this paper, we undertake this problem for nondegenerate manifolds in the context of simultaneous Diophantine approximation. In the past few years, the study of the achievable degrees of freedom in various schemes on Interference Alignment from electronics communication requires one to understand Problem 1.1; see [21,6]. In particular, readers are referred to the works [1,17,13,21,20,22,24], and [6] for more survey.…”

Section: Introductionmentioning

confidence: 99%

“…In the past few years, the study of the achievable degrees of freedom in various schemes on Interference Alignment from electronics communication requires one to understand Problem 1.1; see [21,6]. In particular, readers are referred to the works [1,17,13,21,20,22,24], and [6] for more survey. We expect that our result will have application in this direction.…”

Section: Introductionmentioning

confidence: 99%