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dc.contributor.authorHart, Derrick N.en_US
dc.date.accessioned2006-09-01T19:29:19Z
dc.date.available2006-09-01T19:29:19Z
dc.date.issued2006-06-05en_US
dc.identifier.urihttp://hdl.handle.net/1853/11516
dc.description.abstractRecent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefulness of first studying the problem in a finite field environment. Using the techniques of Bourgain to give a result in other settings such as general abelian groups, the author gives a walk through, including proof, of Roth's theorem in both the one dimensional and two dimensional cases (it would be more accurate to refer to the two dimensional case as Shkredov's Theorem). In the one dimensional case the argument is at its base Meshulam's but the structure will be essentially Green's. Let Ϝⁿ [subscript p], p ≠ 2 be the finite field of cardinality N = pⁿ. For large N, any subset A ⊂ Ϝⁿ [subscript p] of cardinality ∣A ∣≳ N ∕ log N must contain a triple of the form {x, x + d, x + 2d} for x, d ∈ Ϝⁿ [subscript p], d ≠ 0. In the two dimensional case the argument is Lacey and McClain who made considerable refinements to this argument of Green who was bringing the argument to the finite field case from a paper of Shkredov. Let Ϝ ⁿ ₂ be the finite field of cardinality N = 2ⁿ. For all large N, any subset A⊂ Ϝⁿ ₂ × Ϝⁿ ₂ of cardinality ∣A ∣≳ N ² (log n) − [superscript epsilon], ε <, 1, must contain a corner {(x, y), (x + d, y), (x, y + d)} for x, y, d ∈ Ϝⁿ₂ and d ≠ 0.en_US
dc.format.extent223793 bytes
dc.format.mimetypeapplication/pdf
dc.language.isoen_US
dc.publisherGeorgia Institute of Technologyen_US
dc.subjectArithmetic progressionsen_US
dc.subjectCorners
dc.subjectRoth
dc.subjectAdditive combinatorics
dc.titleFinite Field Models of Roth's Theorem in One and Two Dimensionsen_US
dc.typeThesisen_US
dc.description.degreeM.S.en_US
dc.contributor.departmentMathematicsen_US
dc.description.advisorCommittee Chair: Lacey, Michael; Committee Member: Green, Bill; Committee Member: Tetali, Prasaden_US


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