March 19, 2015
Similar papers 2
December 1, 2017
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 based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality a...
February 20, 2009
Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<\alpha|A|$ and $\alpha \ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\mathbb{R}^d$. As an application of the ca...
April 4, 2009
In the present paper we show that if A is a set of n real numbers, and the product set A.A has at most n^(1+c) elements, then the k-fold sumset kA has at least n^(log(k/2)/2 log 2 + 1/2 - f_k(c)) elements, where f_k(c) -> 0 as c -> 0. We believe that the methods in this paper might lead to a much stronger result; indeed, using a result of Trevor Wooley on Vinogradov's Mean Value Theorem and the Tarry-Escott Problem, we show that if |A.A| < n^(1+c), then |k(A.A)| > n^(Omega((k...
January 29, 2015
We give a partial answer to a conjecture of A. Balog, concerning the size of AA+A, where A is a finite subset of real numbers. Also, we prove several new results on the cardinality of A:A+A, AA+AA and A:A + A:A.
October 3, 2020
We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $\delta=|A|^{-\alpha}$, we have: $\mathcal{N}(A+A, \delta)\cdot \mathcal{N}(AA, \delta) \gtrsim_{\epsilon}|A|\delta^{-1+\epsilon}$.
June 29, 2011
In this paper we show that for any $k\geq2$, there exist two universal constants $C_k,D_k>0$, such that for any finite subset $A$ of positive real numbers with $|AA|\leq M|A|$, $|kA|\geq \frac{C_k}{M^{D_k}}\cdot|A|^{\log_42k}.$
February 10, 2021
In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$ This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to also prove the three-variable expans...
February 28, 2024
Given a set of integers $A$ and an integer $k$, write $A+k\cdot A$ for the set $\{a+kb:a\in A,b\in A\}$. Hanson and Petridis showed that if $|A+A|\le K|A|$ then $|A+2\cdot A|\le K^{2.95}|A|$. At a presentation of this result, Petridis stated that the highest known value for $\frac{\log(|A+2\cdot A|/|A|)}{\log(|A+A|/|A|)}$ (bounded above by 2.95) was $\frac{\log 4}{\log 3}$. We show that, for all $\epsilon>0$, there exist $A$ and $K$ with $|A+A|\le K|A|$ but with $|A+2\cdot A|...
March 13, 2018
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 $$\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}.$$ We use similar decompositions to improve upon various sum-product estimates. For instance, we show $$ |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}.$$
August 22, 2011
A set of reals $A=\{a_1,...,a_n\}$ labeled in increasing order is called convex if there exists a continuous strictly convex function $f$ such that $f(i)=a_i$ for every $i$. Given a convex set $A$, we prove \[|A+A|\gg\frac{|A|^{14/9}}{(\log|A|)^{2/9}}.\] Sumsets of different summands and an application to a sum-product-type problem are also studied either as remarks or as theorems.