New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Societ...
| المؤلفون الرئيسيون: | , |
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| التنسيق: | Journal article |
| منشور في: |
2012
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| الملخص: | Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that paper here. Our new argument is different and significantly shorter. In fact we prove a stronger result, which can be viewed as a quantatitive version of some previous results of Bergelson-Host-Kra and the authors. |
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