Title: Regularity and removal lemmas and their applications
Abstract:In this thesis, we analyze the regularity method
pioneered by Szemeredi, and also discuss one of its prevalent
applications, the removal lemma. First, we prove a new lower bound
on the number of parts...In this thesis, we analyze the regularity method
pioneered by Szemeredi, and also discuss one of its prevalent
applications, the removal lemma. First, we prove a new lower bound
on the number of parts required in a version of Szemeredi's
regularity lemma, determining the order of the tower height in that
version up to a constant factor. This addresses a question of
Gowers. Next, we turn to algorithms. We give a fast algorithmic
Frieze-Kannan (weak) regularity lemma that improves on previous
running times. We use this to give a substantially faster
deterministic approximation algorithm for counting subgraphs.
Previously, only an exponential dependence of the running time on
the error parameter was known; we improve it to a polynomial
dependence. We also revisit the problem of finding an algorithmic
regularity lemma, giving approximation algorithms for some
co-NP-complete problems. We show how to use the Frieze-Kannan
regularity lemma to approximate the regularity of a pair of vertex
sets. We also show how to quickly find, for each [epsilon]' >
[epsilon], an [epsilon]'-regular partition with k parts if there
exists an [epsilon]-regular partition with k parts. After studying
algorithms, we turn to the arithmetic setting. Green proved an
arithmetic regularity lemma, and used it to prove an arithmetic
removal lemma. The bounds obtained, however, were tower-type, and
Green posed the problem of improving the quantitative bounds on the
arithmetic triangle removal lemma, and, in particular, asked
whether a polynomial bound holds. The previous best known bound was
tower-type with a logarithmic tower height. We solve Green's
problem, proving an essentially tight bound for Green's arithmetic
triangle removal lemma in Fn/p. Finally, we give a new proof of a
regularity lemma for permutations, improving the previous
tower-type bound on the number of parts to an exponential
bound.Read More
Publication Year: 2017
Publication Date: 2017-01-01
Language: en
Type: dissertation
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