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On an Argument of Shkredov on TwoDimensional Corners
, 2007
"... Let F n 2 be the finite field of cardinality 2 n. For all large n, any subset A ⊂ F n 2 × F n 2 of cardinality A  � 4 n log log n log n, must contain three points {(x, y) , (x + d, y) , (x, y + d)} for x, y, d ∈ F n 2 and d � = 0. Our argument is an elaboration of an argument of Shkredov [14], bu ..."
Abstract

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Let F n 2 be the finite field of cardinality 2 n. For all large n, any subset A ⊂ F n 2 × F n 2 of cardinality A  � 4 n log log n log n, must contain three points {(x, y) , (x + d, y) , (x, y + d)} for x, y, d ∈ F n 2 and d � = 0. Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on log n, which is larger than has been obtained previously.