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Arithmetic progressions : combinatorial and number-theoretic perspectives
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Title
Arithmetic progressions : combinatorial and number-theoretic perspectives
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Vijay
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Sujith
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Sujith Vijay
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author
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Beck
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Jozsef
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Advisory Committee
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Jozsef Beck
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chair
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Advisory Committee
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Endre Szemeredi
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internal member
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Zeilberger
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Doron
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Advisory Committee
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Doron Zeilberger
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internal member
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Gurvich
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Vladimir
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Advisory Committee
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Vladimir Gurvich
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Rutgers University
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Graduate School-New Brunswick
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theses
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2007
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2007
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English
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electronic
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Extent
vi, 59 pages
Abstract
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A beautiful result in the study of arithmetic progressions modulo 1 is the three distance theorem, conjectured by Steinhaus and proved by Sós, Świerczkowski et al. According to this theorem, there are at most three distinct gaps between consecutive elements in any finite initial segment of the sequence of fractional parts of integer multiples of any real number. We interpret this theorem as a statement about the finiteness of the number of champions in a suitably defined tournament, and obtain higher-dimensional generalizations.
A famous open problem in combinatorial discrepancy theory, raised by Erdős many decades ago, is whether the hypergraph of homogeneous arithmetic progressions has unbounded discrepancy. We investigate a variant of this question. In 1986, Beck showed that given any 2-coloring, the hypergraph of quasi-progressions {lfloor n alpha rfloor} corresponding to almost all real numbers α in [1, ∞) has unbounded discrepancy, in fact, at least log* N, the inverse of the tower function. We make a substantial improvement on this lower bound, replacing log* N by (log N) 1/4 - o(1) and also show that there is some quasi-progression with discrepancy at least (1/50) N 1/6.
A fundamental result in Ramsey theory is the theorem of Hales and Jewett, which states that any 2-coloring of the nd hypercube admits a monochromatic line for any fixed n and sufficiently large d. We show that the Hales-Jewett number HJ(n) is at least exponential in n, improving the linear lower bound in the original paper of Hales and Jewett.
We also study a game-theoretic variant of the unbounded discrepancy problem where two players, Maker and Breaker, take turns coloring the integers from 0 to N with their own colors. Maker's goal is to obtain a lead on some homogeneous arithmetic progression that exceeds a pre-specified target, and Breaker's goal is to prevent this from happening. We show that given ε > 0$, Maker wins if the target is below N 1/2 - ε and Breaker wins if the target is above N 1/2 + ε for sufficiently large N.
A famous open problem in combinatorial discrepancy theory, raised by Erdős many decades ago, is whether the hypergraph of homogeneous arithmetic progressions has unbounded discrepancy. We investigate a variant of this question. In 1986, Beck showed that given any 2-coloring, the hypergraph of quasi-progressions {lfloor n alpha rfloor} corresponding to almost all real numbers α in [1, ∞) has unbounded discrepancy, in fact, at least log* N, the inverse of the tower function. We make a substantial improvement on this lower bound, replacing log* N by (log N) 1/4 - o(1) and also show that there is some quasi-progression with discrepancy at least (1/50) N 1/6.
A fundamental result in Ramsey theory is the theorem of Hales and Jewett, which states that any 2-coloring of the nd hypercube admits a monochromatic line for any fixed n and sufficiently large d. We show that the Hales-Jewett number HJ(n) is at least exponential in n, improving the linear lower bound in the original paper of Hales and Jewett.
We also study a game-theoretic variant of the unbounded discrepancy problem where two players, Maker and Breaker, take turns coloring the integers from 0 to N with their own colors. Maker's goal is to obtain a lead on some homogeneous arithmetic progression that exceeds a pre-specified target, and Breaker's goal is to prevent this from happening. We show that given ε > 0$, Maker wins if the target is below N 1/2 - ε and Breaker wins if the target is above N 1/2 + ε for sufficiently large N.
Note
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Ph.D.
Note
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Includes bibliographical references (p. 56-58).
Subject
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Topic
Mathematics
Subject
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Topic
Series, Arithmetic
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.13838
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ETD_152
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doi:10.7282/T30R9PT4
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ETD doctoral
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Open
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Sujith Vijay
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Rutgers University. Graduate School-New Brunswick
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Non-exclusive ETD license
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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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