منابع مشابه
Corrections and Additions to “entropy of Quantum Limits” Jean Bourgain and Elon Lindenstrauss
This theorem implies that every ergodic component of a quantum limit μ under the flow a(t) has entropy ≥ κ′. In a later paper [2], the second named author has been able to use Theorem 1 in conjunction with a partial classification of measures on X invariant under the geodesic flow that satisfy a recurrence property under the Hecke correspondence to prove that the only arithmetic quantum limit i...
متن کاملToward a Unified Theory of Sparse Dimensionality Reduction in Euclidean Space Jean Bourgain and Jelani Nelson
Let Φ ∈ Rm×n be a sparse Johnson-Lindenstrauss transform [KN] with s non-zeroes per column. For T a subset of the unit sphere, ε ∈ (0, 1/2) given, we study settings for m, s required to ensure E Φ sup x∈T ∣∣‖Φx‖22 − 1∣∣ < ε, i.e. so that Φ preserves the norm of every x ∈ T simultaneously and multiplicatively up to 1 + ε. In particular, our most general theorem shows that it suffices to set m = ...
متن کاملBourgain ’ s Theorem
Exercise 1. Show that distances between sets do not necessarily satisfy the triangle inequality. That is, it is possible that d(S1, S2) + d(S2, S3) > d(S1, S3) for some sets S1, S2 and S3. Exercise 2. Prove that d(x, y) ≥ d(S, x)− d(S, y) and thus d(x, y) ≥ |d(S, x)− d(S, y)|. Proof. Fix ε > 0. Let y′ ∈ S be such that d(y′, y) ≤ d(S, y) + ε (if S is a finite set, there is y′ ∈ S s.t. d(y, y′) =...
متن کاملOn a Question of Bombieri and Bourgain
(1) an n+1-tuple (ρ, χ1, ...., χn) of nontrivial C×-valued multiplicative characters of k×, each extended to k by the requirement that it vanish at 0 ∈ k. (2) an n+1-tuple (g, f1, ...., fn) of nonzero one-variable k-polynomials, which are adapted to the character list above in the following sense. Whenever α ∈ k is a zero of g (respectively of some fi), then ρα (respectively χ ordα(fi) i ) is n...
متن کاملThe Bourgain Algebra of a Nest Algebra
In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = algN is a ne...
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ژورنال
عنوان ژورنال: Nature
سال: 2019
ISSN: 0028-0836,1476-4687
DOI: 10.1038/d41586-019-00499-x