Diophantine properties of elements of SO(3)

V. Kaloshin, I. Rodnianski

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12 Scopus citations

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-∈. 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 script I sign = (i1, j1, . . . , im, jm) define |script I sign| = ∑mk=1(|ik|+|jk|) = n and Wn(A, B) = {Wscript I sign(A, B) = Ai1Bj1 . . . AimBjm||script I sign|=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|script I sign|=n ||Wscript I sign(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 sA, B(n) is bounded from below by an exponential function in n2. We also exhibit obstructions to a "simple" proof of the exponential estimate in n.

Original languageEnglish (US)
Pages (from-to)953-970
Number of pages18
JournalGeometric and Functional Analysis
Volume11
Issue number5
DOIs
StatePublished - 2001

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

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