Abstract/Details

Explorations of sum-product phenomena in fields


2010 2010

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Abstract (summary)

We study the sum-product phenomena in finite fields and in reals. The well known sum-product theorem says that under certain circumstances it is not possible for a set to have both an additive and a multiplicative structure. In this dissertation, we first study a quantitative version of the sum-product theorem, and give a stronger sum-product theorem in general fields. We also use Fourier analytic machinery to investigate the expanding property of a family of two variables polynomials. We then use Fourier analytic methods to characterize the polynomials such that when a set behaves like an arithmetic progression, then the size of its image under the polynomials is large. We also study the Erdös-Falconer distance problems in vector space over finite fields. Some analogous results in real setting are also derived.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences; Expanding maps; Finite fields; Products; Sums
Title
Explorations of sum-product phenomena in fields
Author
Shen, Chun-Yen
Number of pages
82
Publication year
2010
Degree date
2010
School code
0093
Source
DAI-B 71/07, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9781124075174
Advisor
Katz, Nets
Committee member
Bercovici, Hari; Demeter, Ciprian; Fisher, David
University/institution
Indiana University
Department
Mathematics
University location
United States -- Indiana
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3409788
ProQuest document ID
725642372
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/725642372
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