The sum-product problem for integers with few prime factors

The sum-product conjecture of Erd\H os and Szemer\'edi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+\delta}$ for every $\delta <1$. In this paper we consider the set of all spectral radii of $n\times n$ matrices with entries in $A$, and find lower bounds for the cardinality of this set. In the case $n=2$, this cardinality is necessarily larger than $\max\{|A+A|,|A\cdot A|\}...

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 $|hA| \ge |A|^{\Omega(e^{\sqrt{c\log{h}}})}$.

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We also establish the inequality $|A+4\cdot A|\geq 5|A|-6 $ for $|A|\geq 5$.

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...

We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y.

Following the sum-product paradigm, we prove that for a set $B$ with polynomial growth, the product set $B.B$ cannot contain large subsets with size of order $|B|^2$ with small doubling. It follows that the additive energy of $B.B$ is asymptotically $o(|B|^6)$. In particular, we extend to sets of small doubling and polynomial growth the classical Multiplication Table theorem of Erd\H{o}s saying that $|[1..n]. [1..n]| = o(n^2)$.

We prove that the size of the product set of any finite arithmetic progression $\mathcal{A}\subset \mathbb{Z}$ satisfies \[|\mathcal A \cdot \mathcal A| \ge \frac{|\mathcal A|^2}{(\log |\mathcal A|)^{2\theta +o(1)} } ,\] where $2\theta=1-(1+\log\log 2)/(\log 2)$ is the constant appearing in the celebrated Erd\H{o}s multiplication table problem. This confirms a conjecture of Elekes and Ruzsa from about two decades ago. If instead $\mathcal{A}$ is relaxed to be a subset of ...

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]

For $A\subseteq \mathbb{R}$, let $A+A=\{a+b: a,b\in A\}$ and $AA=\{ab: a,b\in A\}$. For $k\in \mathbb{N}$, let $SP(k)$ denote the minimum value of $\max\{|A+A|, |AA|\}$ over all $A\subseteq \mathbb{N}$ with $|A|=k$. Here we establish $SP(k)=3k-3$ for $2\leq k \leq 7$, the $k=7$ case achieved for example by $\{1,2,3,4,6,8,12\}$, while $SP(k)=3k-2$ for $k=8,9$, the $k=9$ case achieved for example by $\{1,2,3,4,6,8,9,12,16\}$. For $4\leq k \leq 7$, we provide two proofs using di...

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 ...