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Mat. Sb., 2018, Volume 209, Number 10, Pages 71–88 (Mi msb8966)  

This article is cited in 3 scientific papers (total in 3 papers)

On discrete values of bilinear forms

A. Iosevicha, O. Roche-Newtonb, M. Rudnevc*

a Department of Mathematics, University of Rochester, Rochester, NY, USA
b Johannes Kepler University, Linz, Austria
c Department of Mathematics, University of Bristol, Bristol, UK

Abstract: Let $\omega$ be a nondegenerate skew-symmetric bilinear form in the real plane. We prove that for finite a point set $P\subset \mathbb R^2\setminus\{0\}$, the set $T_\omega(P)$ of nonzero values of $\omega$ on $P\times P$, if nonempty, has cardinality $\Omega(N^{96/137})$.
In the special case when $P=A\times A$, where $A$ is a set of at least two reals, we establish the following sum-product type estimates, corresponding to the symmetric and skew-symmetric form $\omega$:
$$ |AA+ AA|= \Omega(|A|^{19/12}) \quadand\quad |AA-AA|= \Omega( \frac{|A|^{49/32}}{\log^{3/32}|A|}). $$
These estimates improve their basic prototypes $\Omega(N^{2/3})$ and $\Omega(|A|^{3/2})$, which readily follow from the Szemerédi-Trotter theorem.
Bibliography: 28 titles.

Keywords: Erdős problems, sum-product estimates, cross-ratio.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8966

Full text: PDF file (716 kB)
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English version:
Sbornik: Mathematics, 2018, 209:10, 1482–1497

Bibliographic databases:

UDC: 519.1+514.17
MSC: Primary 52C10; Secondary 11B75
Received: 10.05.2017 and 05.08.2017

Citation: A. Iosevich, O. Roche-Newton, M. Rudnev, “On discrete values of bilinear forms”, Mat. Sb., 209:10 (2018), 71–88; Sb. Math., 209:10 (2018), 1482–1497

Citation in format AMSBIB
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\by A.~Iosevich, O.~Roche-Newton, M.~Rudnev
\paper On discrete values of bilinear forms
\jour Mat. Sb.
\yr 2018
\vol 209
\issue 10
\pages 71--88
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\crossref{https://doi.org/10.4213/sm8966}
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\elib{https://elibrary.ru/item.asp?id=35601304}
\transl
\jour Sb. Math.
\yr 2018
\vol 209
\issue 10
\pages 1482--1497
\crossref{https://doi.org/10.1070/SM8966}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85059157642}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. B. Murphy, G. Petridis, “Products of difference over arbitrary finite fields”, Discrete Anal., 2018, 18, 42 pp.  crossref  mathscinet  zmath  isi
    2. 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
    3. M. Rudnev, G. Shakan, I. D. Shkredov, “Stronger sum-product inequalities for small sets”, Proc. Amer. Math. Soc., 148:4 (2020), 1467–1479  crossref  mathscinet  zmath  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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