# New results on sum-products in R

@article{Konyagin2016NewRO, title={New results on sum-products in R}, author={Sergei Konyagin and Ilya D. Shkredov}, journal={arXiv: Combinatorics}, year={2016} }

We improve a previous sum--products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure energy sum--products results, improving a result of Balog and Wooley, in particular.

#### 23 Citations

Some remarks on the asymmetric sum-product phenomenon

- Mathematics, Physics
- Moscow Journal of Combinatorics and Number Theory
- 2019

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max\{|AB|, |A+C| \}$ and $\max\{|(A+\alpha)B|, |A+C|\}$, $\alpha \neq 0$ in the regime when the… Expand

An application of the sum-product phenomenon to sets having no solutions of several linear equations

- Mathematics
- 2016

We prove that for an arbitrary $\kappa \le \frac{1}{3}$ any subset of $\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^\kappa)$. We also find several… Expand

On the few products, many sums problem

- Mathematics
- Journal de Théorie des Nombres de Bordeaux
- 2019

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime… Expand

N T ] 1 6 Ju l 2 01 8 Sum-product estimates over arbitrary finite fields

- 2018

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets A ⊂ Fq we have |(A−A) + (A−A)| ≫ |A| 1 21 . This… Expand

Some remarks on the Balog–Wooley decomposition theorem and quantities D+, D×

- Mathematics
- 2016

In the paper we study two characteristics D^+ (A), D^\times (A) of a set A which play important role in recent results concerning sum-product phenomenon. Also we obtain several variants and… Expand

Breaking the 6/5 threshold for sums and products modulo a prime

- Mathematics
- 2018

Let $A \subset \mathbb{F}_p$ of size at most $p^{3/5}$. We show $$|A+A| + |AA| \gtrsim |A|^{6/5 + c},$$ for $c = 4/305$. Our main tools are the cartesian product point--line incidence theorem of… Expand

On the energy variant of the sum-product conjecture

- Mathematics
- Revista Matemática Iberoamericana
- 2019

We prove new exponents for the energy version of the Erdős-Szemeredi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard… Expand

Difference sets are not multiplicatively closed

- Mathematics
- 2016

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar… Expand

Some remarks on sets with small quotient set

- Mathematics
- 2016

We prove, in particular, that for any finite set of real numbers A with |A/A| \ll |A| one has |A-A| > |A|^{5/3 - o(1)}. Also we show that |3A| > |A|^{2-o(1)} in the case.

An update on the sum-product problem

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2021

We improve the best known sum-product estimates over the reals. We prove that
\[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\]
for a finite
$A\subset \mathbb… Expand

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