On the local structure of a generic central set
Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups
This work concerns topological spaces of the following types: open subsets of normed vector spaces, manifolds over normed vector spaces, the closures of open subsets of normed vector spaces and some other types of topological spaces related to the above. We show that such spaces are determined by various subgroups of their auto-homeomorphism groups. Theorems 1-3 below are typical examples of the results obtained in this work. Theorem 1. For a metric space X let UC(X) denote the group of all auto-homeomorphisms...
An observation on the Turán-Nazarov inequality
The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter the original Turán-Nazarov inequality, they necessarily enter the definition of ω.
Bernstein classes
One of the classical Bernstein inequalities compares the maxima of a polynomial of a given degree on the interval [-1,1] and on the ellipse in the complex plane with the focuses -1, 1 and the semiaxes . We prove a similar inequality for a branch of an algebraic function of a given degree on the maximal disk of its regularity, with the explicitly given constant, depending on the degree only. In particular, this improves a recent inequality of Fefferman and Narasimhan and answers one of their questions....
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