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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 1, Pages 161–182 (Mi izv1140)  

This article is cited in 15 scientific papers (total in 15 papers)

On sets of large trigonometric sums

I. D. Shkredov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove the existence of non-trivial solutions of the equation $r_1+r_2=r_3+r_4$, where $r_1$, $r_2$, $r_3$ and $r_4$ belong to the set $R$ of large Fourier coefficients of a certain subset $A$ of $\mathbb Z/N\mathbb Z$. This implies that $R$ has strong additive properties. We discuss generalizations and applications of the results obtained.

DOI: https://doi.org/10.4213/im1140

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English version:
Izvestiya: Mathematics, 2008, 72:1, 149–168

Bibliographic databases:

UDC: 511.218+511.336
MSC: 11B25, 05D10, 11L07, 37C25, 37B20, 37A05, 28D05
Received: 12.07.2006

Citation: I. D. Shkredov, “On sets of large trigonometric sums”, Izv. RAN. Ser. Mat., 72:1 (2008), 161–182; Izv. Math., 72:1 (2008), 149–168

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. D. Shkredov, “Fourier analysis in combinatorial number theory”, Russian Math. Surveys, 65:3 (2010), 513–567  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Konyagin S.V., Shkredov I.D., “On one result of J. Bourgain”, Ukrainian Math. J., 62:3 (2010), 380–419  crossref  mathscinet  zmath  isi  scopus
    3. Shkredov I.D., Yekhanin S., “Sets with large additive energy and symmetric sets”, J. Combin. Theory Ser. A, 118:3 (2011), 1086–1093  crossref  mathscinet  zmath  isi  elib  scopus
    4. Sanders T., “A quantitative version of the non-abelian idempotent theorem”, Geom. Funct. Anal., 21:1 (2011), 141–221  crossref  mathscinet  zmath  isi  elib  scopus
    5. Sanders T., “On the Bogolyubov–Ruzsa lemma”, Anal. PDE, 5:3 (2012), 627–655  crossref  mathscinet  zmath  isi  elib  scopus
    6. Bateman M., Katz N.H., “New bounds on cap sets”, J. Amer. Math. Soc., 25:2 (2012), 585–613  crossref  mathscinet  zmath  isi  scopus
    7. Sanders T., “On certain other sets of integers”, J. Anal. Math., 116 (2012), 53–82  crossref  mathscinet  zmath  isi  elib  scopus
    8. Sanders T., “Structure in Sets with Logarithmic Doubling”, Can. Math. Bul.-Bul. Can. Math., 56:2 (2013), 412–423  crossref  mathscinet  zmath  isi  scopus
    9. Schoen T., Shkredov I.D., “Higher Moments of Convolutions”, J. Number Theory, 133:5 (2013), 1693–1737  crossref  mathscinet  zmath  isi  elib  scopus
    10. I. D. Shkredov, “Structure theorems in additive combinatorics”, Russian Math. Surveys, 70:1 (2015), 113–163  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. S. V. Konyagin, I. D. Shkredov, “A quantitative version of the Beurling-Helson theorem”, Funct. Anal. Appl., 49:2 (2015), 110–121  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. Bloom T.F., “A quantitative improvement for Roth's theorem on arithmetic progressions: Table 1.”, J. Lond. Math. Soc.-Second Ser., 93:3 (2016), 643–663  crossref  mathscinet  zmath  isi  scopus
    13. Aggarwal D., Hosseini K., Lovett Sh., “Affine-Malleable Extractors, Spectrum Doubling, and Application to Privacy Amplification”, 2016 IEEE International Symposium on Information Theory, IEEE International Symposium on Information Theory, IEEE, 2016, 2913–2917  isi
    14. I. D. Shkredov, “An application of the sum-product phenomenon to sets avoiding several linear equations”, Sb. Math., 209:4 (2018), 580–603  mathnet  crossref  crossref  adsnasa  isi  elib
    15. I. D. Shkredov, “Korotkoe zamechanie o multiplikativnoi energii spektra”, Matem. zametki, 105:3 (2019), 444–454  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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