Type of Document	Master's Thesis
Author	Hart, Derrick Norman
URN	etd-05292006-103834
Title	Finite Field Models of Roth's Theorem in One and Two Dimensions
Degree	Master of Science
Department	Mathematics
Advisory Committee	Advisor Name Title Lacey, Michael Committee Chair Green, Bill Committee Member Tetali, Prasad Committee Member
Keywords	arithmetic progressions corners Roth additive combinatorics
Date of Defense	2006-05-31
Availability	unrestricted
Abstract Recent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefullness of first studying the problem in a finite field enviroment. Then 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 $F_p^n,p eq 2$ be the finite field of cardinality $N=p^n$. For large N, any subset $A subset F_p^n$ of cardinality$$|A|gtrsim frac{N}{log N}$$ must contain a triple of the form ${x,x+d,x+2d}$ for $x,d in F_p^n, d eq 0$. 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 $mathbb F _2^n$ be the finite field of cardinality $N=2 ^{n}$. For all large $N$, any subset $Asubset mathbb F _2^n imes mathbb F _2 ^n$ of cardinality egin{equation*} abs{ A} gtrsim N^2 (log n) ^{-epsilon },,qquad epsilon <,1,, end{equation*} must contain a corner $ {(x,y),,(x+d,y),,(x,y+d)}$ for $x,y,din mathbb F_2^n$ and $d eq 0$.
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