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2009•CaltechTHESIS (California Institute of Technology)Open access

Boundaries of Smooth Sets and Singular Sets of Blaschke Products in the Little Bloch Class

Hungerford, Gregory Jude

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Abstract

A subset of R is called smooth if the integral of its characteristic function is smooth in the sense defined by Zygmund. It is shown that such a set is either trivial or its boundary has Hausdorff dimension 1. Sets are constructed here which are as close to smooth as one likes but whose boundaries do not have dimension 1. It was conjectured by T. Wolff that if B is Blaschke product in the Little Bloch class, its zeroes accumulate to a set of dimension 1. This conjecture is proven here.

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A subset of R is called smooth if the integral of its characteristic function is smooth in the sense defined by Zygmund. It is shown that such a set is either trivial or its boundary has Hausdorff dimension 1. Sets are constructed here which are as close to smooth as one likes but whose boundaries do not have dimension 1. It was conjectured by T. Wolff that if B is Blaschke product in the Little Bloch class, its zeroes accumulate to a set of dimension 1. This conjecture is proven here.

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Available abstract

A subset of R is called smooth if the integral of its characteristic function is smooth in the sense defined by Zygmund. It is shown that such a set is either trivial or its boundary has Hausdorff dimension 1. Sets are constructed here which are as close to smooth as one likes but whose boundaries do not have dimension 1. It was conjectured by T. Wolff that if B is Blaschke product in the Little Bloch class, its zeroes accumulate to a set of dimension 1. This conjecture is proven here.

Key concepts: Blaschke product, Mathematics, Conjecture, Hausdorff dimension, Boundary (topology), Class (philosophy), Dimension (graph theory), Packing dimension

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