Variations on the Sum-Product Problem

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...

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...

The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\R$ and cyclic groups $\Z/q\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain ...

Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof of this bound is a new superquadratic expander involving products and shifts. We prove that, for any finite set $X \subset \mathbb R$, there exist $z,z' \in X$ such that \[ \left|\frac{(zX+1)^{(2)}(z'X+1)^{(2)}}{(zX+1)^{(2)}(z'X+1)}\right| \...

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim |A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim \frac{{|A|}^{{3/2}}}{q^{{1/4}}$.

In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if $A\subseteq \mathbb{F}_p$ satisfies $|A|\le p^{64/117}$ then $$ \max\{|A\pm A|, |AA|\} \gtrsim |A|^{39/32}. $$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estim...

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 main result is the bound $$\left| \frac{A+A}{A+A} \right| \gg \frac{|A|^{2+\frac{2}{25}}}{|A:A|^{\frac{1}{25}}\log |A|},$$ where $A:A$ denotes the ratio set of $A$. This improves on a result of Balog and the author (arXiv:1402.5775), provid...

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present paper we prove that if $A\subset \mathbb{F}_p$ with $|A|>p^{2/3},$ then $$ \max\{|A+A|, |AA|\}\gg p^{1/2}|A|^{1/2}. $$

In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $\Delta(\alpha)>0$ such that if $|A|\asymp p^{\alpha}$ then $$ \max(|A+A|,|A\cdot A|)\gg |A|^{1+\Delta(\alpha)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g...

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 $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to $$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \gg |A|^{73...