On an Argument of Shkredov on Two-Dimensional Corners
نویسندگان
چکیده
Let F2 be the finite field of cardinality 2 . For all large n, any subset A ⊂ F2 × F n 2 of cardinality |A| & 4 log log n log n , must contain three points {(x, y) , (x + d, y) , (x, y + d)} for x, y, d ∈ F2 and d 6= 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.
منابع مشابه
On a Two–dimensional Analog of Szemerédi’s Theorem in Abelian Groups
Let G be a finite Abelian group and A ⊆ G×G be a set of cardinality at least |G|/(log log |G|)c, where c > 0 is an absolute constant. We prove that A contains a triple {(k, m), (k + d, m), (k, m+ d)}, where d 6= 0. This theorem is a two-dimensional generalization of Szemerédi’s theorem on arithmetic progressions.
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