An inverse theorem for the Gowers U^3 norm
The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm...
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| Materialtyp: | Journal article |
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2005
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| Sammanfattning: | The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm, obtaining a reasonably complete description of functions f : G -> C for which ||f||_{U^3} is large and providing links to recent results of Host, Kra and Ziegler in ergodic theory. As an application we generalise a result of Gowers on Szemeredi's theorem. Writing r_4(G) for the size of the largest set A not containing four distinct elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c} for some absolute constant c. In future papers we will develop these ideas further, obtaining an asymptotic for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as well as superior bounds for r_4(G). |
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