Related papers: New results on sum-products in R

We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved…

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a…

Let $A \subset \mathbb{R}$ be finite. We quantitatively improve the Balog-Wooley decomposition, that is $A$ can be partitioned into sets $B$ and $C$ such that $$\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) ,…

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

In this paper, we consider the sum-product problem of obtaining lower bounds for the size of the set $$\frac{A+A}{A+A}:=\left \{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right\},$$ for an arbitrary finite set $A$ of real numbers. The…

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A…

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set…

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

We prove that the sumset or the productset of any finite set of real numbers, $A,$ is at least $|A|^{4/3-\epsilon},$ improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, $E(A,A).$

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 sizes of finite subsets $A,B,C$ of a field…

New lower bounds involving sum, difference, product, and ratio sets for $A\subset \C$ are given.

In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}-o(1)} \] holds for all $A \subset \mathbb R$, and for all convex functions $f$ which satisfy an additional technical condition.…

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 residue field. The improvements are…

Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this paper, we improve a recent result due to Yazici (2018) on a sum-product type problem.…

In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool…

This paper gives an improved sum-product estimate for subsets of a finite field whose order is not prime. It is shown, under certain conditions, that $$\max\{|A+A|,|A\cdot{A}|\}\gg{\frac{|A|^{12/11}}{(\log_2|A|)^{5/11}}}.$$ This new…

Given a subset of real numbers $A$ with small product $AA$ we obtain a new upper bound for the additive energy of $A$. The proof uses a natural observation that level sets of convolutions of the characteristic function of $A$ have small…

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each…

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\;…

We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and…