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Mat. Sb., 2017, Volume 208, Number 12, Pages 144–158 (Mi msb8733)  

This article is cited in 6 scientific papers (total in 6 papers)

Some remarks on sets with small quotient set

I. D. Shkredov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We prove that for any finite set $A\subset \mathbb{R}$ with $|A/A| \ll |A|$ we have $|A-A| \gg |A|^{5/3 - o(1)}$. We also show that for such sets $|A+A+A| \gg |A|^{2-o(1)}$.
Bibliography: 22 titles.

Keywords: additive combinatorics, sum-product phenomenon, Erdős-Szemerédi conjecture.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Science Foundation under grant no. 14-50-00005.


DOI: https://doi.org/10.4213/sm8733

Full text: PDF file (920 kB)
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English version:
Sbornik: Mathematics, 2017, 208:12, 1854–1868

Bibliographic databases:

UDC: 511.218
MSC: 11B75
Received: 16.05.2016 and 16.09.2016

Citation: I. D. Shkredov, “Some remarks on sets with small quotient set”, Mat. Sb., 208:12 (2017), 144–158; Sb. Math., 208:12 (2017), 1854–1868

Citation in format AMSBIB
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  • https://doi.org/10.4213/sm8733
  • http://mi.mathnet.ru/eng/msb/v208/i12/p144

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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. B. Murphy, O. Roche-Newton, I. D. Shkredov, “Variations on the sum-product problem II”, SIAM Discret. Math., 31:3 (2017), 1878–1894  mathnet  crossref  mathscinet  zmath  isi  scopus
    2. A. Iosevich, O. Roche-Newton, M. Rudnev, “On discrete values of bilinear forms”, Sb. Math., 209:10 (2018), 1482–1497  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. B. Hanson, O. Roche-Newton, D. Zhelezov, “On iterated product sets with shifts”, Mathematika, 65:4 (2019), 831–850  crossref  mathscinet  zmath  isi  scopus
    4. B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev, I. D. Shkredov, “New results on sum-product type growth over fields”, Mathematika, 65:3 (2019), 588–642  crossref  mathscinet  zmath  isi  scopus
    5. B. Murphy, M. Rudnev, I. D. Shkredov, Yu. N. Shteinikov, “On the few products, many sums problem”, J. Theor. Nr. Bordx., 31:3 (2019), 573–602  mathscinet  zmath  isi
    6. K. I. Olmezov, A. S. Semchankau, I. D. Shkredov, “On Popular Sums and Differences for Sets with Small Multiplicative Doubling”, Math. Notes, 108:4 (2020), 557–565  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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