### Abstract

It is established that there exists an absolute constant c > 0 such that for any finite set A of positive real numbers |AA + A| >> |A|^{3/2+c} On the other hand, we give an explicit construction of a finite set A ⊂ R such that |AA + A| = o(|A|^{2}), disproving a conjecture of Balog.

Original language | English |
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Journal | Journal of the London Mathematical Society |

DOIs | |

Publication status | Accepted/In press - Jan 1 2018 |

### Fingerprint

### Keywords

- 11B30 (secondary)
- 52C10 (primary)

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the London Mathematical Society*. https://doi.org/10.1112/jlms.12177

**On the size of the set AA+A.** / Roche-Newton, Oliver; Ruzsa, I.; Shen, Chun Yen; Shkredov, Ilya D.

Research output: Contribution to journal › Article

*Journal of the London Mathematical Society*. https://doi.org/10.1112/jlms.12177

}

TY - JOUR

T1 - On the size of the set AA+A

AU - Roche-Newton, Oliver

AU - Ruzsa, I.

AU - Shen, Chun Yen

AU - Shkredov, Ilya D.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - It is established that there exists an absolute constant c > 0 such that for any finite set A of positive real numbers |AA + A| >> |A|3/2+c On the other hand, we give an explicit construction of a finite set A ⊂ R such that |AA + A| = o(|A|2), disproving a conjecture of Balog.

AB - It is established that there exists an absolute constant c > 0 such that for any finite set A of positive real numbers |AA + A| >> |A|3/2+c On the other hand, we give an explicit construction of a finite set A ⊂ R such that |AA + A| = o(|A|2), disproving a conjecture of Balog.

KW - 11B30 (secondary)

KW - 52C10 (primary)

UR - http://www.scopus.com/inward/record.url?scp=85048971513&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048971513&partnerID=8YFLogxK

U2 - 10.1112/jlms.12177

DO - 10.1112/jlms.12177

M3 - Article

AN - SCOPUS:85048971513

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

ER -