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Some applications of Freiman's inverse theorem

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TypeOfResource
Text
TitleInfo (ID = T-1)
Title
Some applications of Freiman's inverse theorem
Identifier
ETD_2563
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000053124
Language
LanguageTerm (authority = ISO639-2); (type = code)
eng
Genre (authority = marcgt)
theses
Subject (ID = SBJ-1)
Name (authority = LC-NAF)
NamePart (type = personal)
Freiman, G. A.
Subject (ID = SBJ-2); (authority = RUETD)
Topic
Mathematics
Subject (ID = SBJ-3); (authority = ETD-LCSH)
Topic
Additive combinatorics
Subject (ID = SBJ-4); (authority = ETD-LCSH)
Topic
Number theory
Abstract (type = abstract)
The celebrated Freiman's inverse theorem in Additive Combinatorics asserts that an additive set of small doubling constant must have additive structure. This thesis contains two applications achieved by combining this theorem with a dyadic pigeonhole principle technique. 1. A finite set A of integers is square-sum-free if no subset of A sums up to a square. In 1986, Erdos posed the problem of determining the largest cardinality of a square-sum-free subset of {1,..., n}. Significantly improving earlier results, we show in Chapter 2 that this maximum cardinality is of order n^{1/3+o(1)}, which is asymptotically tight. 2. A classical result of Littlewood-Offord and Erdos from the 1940s asserts that if the v_i are non-zero, then the concentration probability of the (multi)set V={v_1,...,v_n},
ho(V) := sup_{x} P( v_1 eta_1+ ... + v_n eta_n=x), is of order O(n^{-1/2}), where eta_i are i.i.d. copies of a Bernoulli random variable. Motivated by problems concerning random matrices, Tao and Vu introduced the Inverse Littlewood-Offord problem. In the inverse problem, one would like to give a characterization of the set V, given that ho(V) is relatively large. In Chapter 3, we develop a method to attack the inverse problem. As an application, we strengthen several previous results of Tao and Vu, obtaining an almost optimal characterization for $V$. This implies several classical theorems, such as those of Sarkozy-Szemeredi, Halasz, and Stanley.
PhysicalDescription
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electronic resource
Extent
vii, 80 p.
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application/pdf
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Note (type = degree)
Ph.D.
Note
Includes abstract
Note
Vita
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Hoi H. Nguyen
Name (ID = NAME-1); (type = personal)
NamePart (type = family)
Nguyen
NamePart (type = given)
Hoi H.
NamePart (type = date)
1980-
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author
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Hoi Nguyen
Name (ID = NAME-2); (type = personal)
NamePart (type = family)
Vu
NamePart (type = given)
Van
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chair
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Advisory Committee
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Van Vu
Name (ID = NAME-3); (type = personal)
NamePart (type = family)
Kahn
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Jeff
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internal member
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Advisory Committee
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Jeff Kahn
Name (ID = NAME-4); (type = personal)
NamePart (type = family)
Szemeredi
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Endre
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internal member
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Advisory Committee
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Endre Szemeredi
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Pemantle
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Robin
Role
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outside member
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Advisory Committee
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Robin Pemantle
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NamePart
Rutgers University
Role
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degree grantor
Name (ID = NAME-2); (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
OriginInfo
DateCreated (qualifier = exact)
2010
DateOther (qualifier = exact); (type = degree)
2010
Place
PlaceTerm (type = code)
xx
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3S46S1V
Genre (authority = ExL-Esploro)
ETD doctoral
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RightsDeclaration (AUTHORITY = GS); (ID = rulibRdec0006)
The author owns the copyright to this work.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
RightsHolder (ID = PRH-1); (type = personal)
Name
FamilyName
Nguyen
GivenName
Hoi
Role
Copyright Holder
RightsEvent (ID = RE-1); (AUTHORITY = rulib)
Type
Permission or license
DateTime
2010-04-09 15:04:12
AssociatedEntity (ID = AE-1); (AUTHORITY = rulib)
Role
Copyright holder
Name
Hoi Nguyen
Affiliation
Rutgers University. Graduate School - New Brunswick
AssociatedObject (ID = AO-1); (AUTHORITY = rulib)
Type
License
Name
Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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ETD
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application/pdf
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application/x-tar
FileSize (UNIT = bytes)
552960
Checksum (METHOD = SHA1)
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