---
_id: '8422'
abstract:
- lang: eng
  text: 'The Birkhoff conjecture says that the boundary of a strictly convex integrable
    billiard table is necessarily an ellipse. In this article, we consider a stronger
    notion of integrability, namely integrability close to the boundary, and prove
    a local version of this conjecture: a small perturbation of an ellipse of small
    eccentricity which preserves integrability near the boundary, is itself an ellipse.
    This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability
    was assumed on a larger set. In particular, it shows that (local) integrability
    near the boundary implies global integrability. One of the crucial ideas in the
    proof consists in analyzing Taylor expansion of the corresponding action-angle
    coordinates with respect to the eccentricity parameter, deriving and studying
    higher order conditions for the preservation of integrable rational caustics.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Guan
  full_name: Huang, Guan
  last_name: Huang
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Alfonso
  full_name: Sorrentino, Alfonso
  last_name: Sorrentino
citation:
  ama: Huang G, Kaloshin V, Sorrentino A. Nearly circular domains which are integrable
    close to the boundary are ellipses. <i>Geometric and Functional Analysis</i>.
    2018;28(2):334-392. doi:<a href="https://doi.org/10.1007/s00039-018-0440-4">10.1007/s00039-018-0440-4</a>
  apa: Huang, G., Kaloshin, V., &#38; Sorrentino, A. (2018). Nearly circular domains
    which are integrable close to the boundary are ellipses. <i>Geometric and Functional
    Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s00039-018-0440-4">https://doi.org/10.1007/s00039-018-0440-4</a>
  chicago: Huang, Guan, Vadim Kaloshin, and Alfonso Sorrentino. “Nearly Circular Domains
    Which Are Integrable Close to the Boundary Are Ellipses.” <i>Geometric and Functional
    Analysis</i>. Springer Nature, 2018. <a href="https://doi.org/10.1007/s00039-018-0440-4">https://doi.org/10.1007/s00039-018-0440-4</a>.
  ieee: G. Huang, V. Kaloshin, and A. Sorrentino, “Nearly circular domains which are
    integrable close to the boundary are ellipses,” <i>Geometric and Functional Analysis</i>,
    vol. 28, no. 2. Springer Nature, pp. 334–392, 2018.
  ista: Huang G, Kaloshin V, Sorrentino A. 2018. Nearly circular domains which are
    integrable close to the boundary are ellipses. Geometric and Functional Analysis.
    28(2), 334–392.
  mla: Huang, Guan, et al. “Nearly Circular Domains Which Are Integrable Close to
    the Boundary Are Ellipses.” <i>Geometric and Functional Analysis</i>, vol. 28,
    no. 2, Springer Nature, 2018, pp. 334–92, doi:<a href="https://doi.org/10.1007/s00039-018-0440-4">10.1007/s00039-018-0440-4</a>.
  short: G. Huang, V. Kaloshin, A. Sorrentino, Geometric and Functional Analysis 28
    (2018) 334–392.
date_created: 2020-09-17T10:42:30Z
date_published: 2018-03-18T00:00:00Z
date_updated: 2021-01-12T08:19:11Z
day: '18'
doi: 10.1007/s00039-018-0440-4
extern: '1'
external_id:
  arxiv:
  - '1705.10601'
intvolume: '        28'
issue: '2'
keyword:
- Geometry and Topology
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1705.10601
month: '03'
oa: 1
oa_version: Preprint
page: 334-392
publication: Geometric and Functional Analysis
publication_identifier:
  issn:
  - 1016-443X
  - 1420-8970
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Nearly circular domains which are integrable close to the boundary are ellipses
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 28
year: '2018'
...
---
_id: '8524'
abstract:
- lang: eng
  text: 'A number α∈R is diophantine if it is not well approximable by rationals,
    i.e. for some C,ε>0 and any relatively prime p,q∈Z we have |αq−p|>Cq−1−ε. It is
    well-known and is easy to prove that almost every α in R is diophantine. In this
    paper we address a noncommutative version of the diophantine properties. Consider
    a pair A,B∈SO(3) and for each n∈Z+ take all possible words in A, A -1, B, and
    B - 1 of length n, i.e. for a multiindex I=(i1,i1,…,im,jm) define |I|=∑mk=1(|ik|+|jk|)=n
    and \( W_n(A,B ) = \{W_{\cal I}(A,B) = A^{i_1} B^{j_1} \dots A^{i_m} B^{j_m}\}_{|{\cal
    I|}=n \).¶Gamburd—Jakobson—Sarnak [GJS] raised the problem: prove that for Haar
    almost every pair A,B∈SO(3) the closest distance of words of length n to the identity,
    i.e. sA,B(n)=min|I|=n∥WI(A,B)−E∥, is bounded from below by an exponential function
    in n. This is the analog of the diophantine property for elements of SO(3). In
    this paper we prove that s A,B (n) is bounded from below by an exponential function
    in n 2. We also exhibit obstructions to a “simple” proof of the exponential estimate
    in n.'
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: I.
  full_name: Rodnianski, I.
  last_name: Rodnianski
citation:
  ama: Kaloshin V, Rodnianski I. Diophantine properties of elements of SO(3). <i>Geometric
    And Functional Analysis</i>. 2001;11(5):953-970. doi:<a href="https://doi.org/10.1007/s00039-001-8222-8">10.1007/s00039-001-8222-8</a>
  apa: Kaloshin, V., &#38; Rodnianski, I. (2001). Diophantine properties of elements
    of SO(3). <i>Geometric And Functional Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s00039-001-8222-8">https://doi.org/10.1007/s00039-001-8222-8</a>
  chicago: Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements
    of SO(3).” <i>Geometric And Functional Analysis</i>. Springer Nature, 2001. <a
    href="https://doi.org/10.1007/s00039-001-8222-8">https://doi.org/10.1007/s00039-001-8222-8</a>.
  ieee: V. Kaloshin and I. Rodnianski, “Diophantine properties of elements of SO(3),”
    <i>Geometric And Functional Analysis</i>, vol. 11, no. 5. Springer Nature, pp.
    953–970, 2001.
  ista: Kaloshin V, Rodnianski I. 2001. Diophantine properties of elements of SO(3).
    Geometric And Functional Analysis. 11(5), 953–970.
  mla: Kaloshin, Vadim, and I. Rodnianski. “Diophantine Properties of Elements of
    SO(3).” <i>Geometric And Functional Analysis</i>, vol. 11, no. 5, Springer Nature,
    2001, pp. 953–70, doi:<a href="https://doi.org/10.1007/s00039-001-8222-8">10.1007/s00039-001-8222-8</a>.
  short: V. Kaloshin, I. Rodnianski, Geometric And Functional Analysis 11 (2001) 953–970.
date_created: 2020-09-18T10:50:11Z
date_published: 2001-12-01T00:00:00Z
date_updated: 2021-01-12T08:19:52Z
day: '01'
doi: 10.1007/s00039-001-8222-8
extern: '1'
intvolume: '        11'
issue: '5'
language:
- iso: eng
month: '12'
oa_version: None
page: 953-970
publication: Geometric And Functional Analysis
publication_identifier:
  issn:
  - 1016-443X
  - 1420-8970
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Diophantine properties of elements of SO(3)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2001'
...