February 12, 2015
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September 3, 2003
We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erd\"os and Szemer\'edi on sum and product sets.
September 25, 2014
Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $\epsilon > 0$ such that if $A \subseteq \mathbb{R}$ is sufficiently large and $|A.A| \le |A|^{1+\epsilon}$, then $|hA| \ge |A|^{\Omega(e^{\sqrt{c\log{h}}})}$.
February 19, 2015
In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s dist...
May 15, 2022
We develop the theory of the additive dimension ${\rm dim} (A)$, i.e. the size of a maximal dissociated subset of a set $A$. It was shown that the additive dimension is closely connected with the growth of higher sumsets $nA$ of our set $A$. We apply this approach to demonstrate that for any small multiplicative subgroup $\Gamma$ the sequence $|n\Gamma|$ grows very fast. Also, we obtain a series of applications to the sum--product phenomenon and to the Balog--Wooley decomposi...
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: $\mathcal{N}(A+A, \delta)\cdot \mathcal{N}(AA, \delta) \gtrsim_{\epsilon}|A|\delta^{-1+\epsilon}$.
April 24, 2012
We are discussing the theorem about the volume of a set $A$ of $Z^n$ having a small doubling property $|2A| < Ck, k=|A|$ and oher problems of Structure Theory of Set Addition (Additive Combinatorics).
March 24, 2010
We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $\Lambda...
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|...
October 5, 2014
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 is a new result which provides a nontrivial upper bound on the multiplicative energy of a sum set or difference set.
June 5, 2018
The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\{a_1\cdots a_k : a_1, \dots, a_k \in A \}$. Furthermore, our method of proof also gives the following $l_{\infty}$ sum-product estimate. For all $\gamma >0$ there exists a constant $C=C(\gamma...