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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 ..."
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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.
The
"... dichotomy between structure and randomness, arithmetic progressions, and the primes ..."
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dichotomy between structure and randomness, arithmetic progressions, and the primes
1 A New Sifting function 1 ()nJ ω+ in
"... We define that prime equations 1 1 1 ( , ,), , ( ,)n k nf P P f P PL L L （5） are polynomials (with integer coefficients) irreducible over integers, where 1, , nP PL are all prime. If sifting function 1 ( ) 0nJ ω+ = then （5）has finite prime solutions. If 1 ( ) 0nJ ω+ ≠ then there are infin ..."
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We define that prime equations 1 1 1 ( , ,), , ( ,)n k nf P P f P PL L L （5） are polynomials (with integer coefficients) irreducible over integers, where 1, , nP PL are all prime. If sifting function 1 ( ) 0nJ ω+ = then （5）has finite prime solutions. If 1 ( ) 0nJ ω+ ≠ then there are infinitely many primes 1, , nP PL such that 1, kf fL are primes. We obtain a unite prime formula in prime distribution primes}are,,:,,{)1, ( 111 kffNPPnN knk LL ≤=++π