@article{8422,
  abstract     = {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.},
  author       = {Huang, Guan and Kaloshin, Vadim and Sorrentino, Alfonso},
  issn         = {1016-443X},
  journal      = {Geometric and Functional Analysis},
  keywords     = {Geometry and Topology, Analysis},
  number       = {2},
  pages        = {334--392},
  publisher    = {Springer Nature},
  title        = {{Nearly circular domains which are integrable close to the boundary are ellipses}},
  doi          = {10.1007/s00039-018-0440-4},
  volume       = {28},
  year         = {2018},
}

@article{8524,
  abstract     = {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.},
  author       = {Kaloshin, Vadim and Rodnianski, I.},
  issn         = {1016-443X},
  journal      = {Geometric And Functional Analysis},
  number       = {5},
  pages        = {953--970},
  publisher    = {Springer Nature},
  title        = {{Diophantine properties of elements of SO(3)}},
  doi          = {10.1007/s00039-001-8222-8},
  volume       = {11},
  year         = {2001},
}