Related papers: Variations on the sum-product problem II

Let $A$ and $B$ be additive sets of $\mathbb{Z}_{2k}$, where $A$ has cardinality $k$ and $B=v.\complement A$ with $v\in\mathbb{Z}_{2k}^{\times}$. In this note some bounds for the cardinality of $A+B$ are obtained, using four different…

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…

We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\subset \mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\asymp \log\lvert A\rvert$) such that \[\max(\lvert…

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such…

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} \;\;…

This is a sequel to arXiv:1308.3604. We study applications to limit multiplicity generalizing the results of arXiv:1208.2257.

In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also…

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

We classify the polynomials $f(x,y) \in \mathbb R[x,y]$ such that given any finite set $A \subset \mathbb R$ if $|A+A|$ is small, then $|f(A,A)|$ is large. In particular, the following bound holds : $|A+A||f(A,A)| \gtrsim |A|^{5/2}.$ The…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

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…

Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$.

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where…

Given $d,s \in \mathbb{N}$, a finite set $A \subseteq \mathbb{Z}$ and polynomials $\varphi_1, \dots, \varphi_{s} \in \mathbb{Z}[x]$ such that $1 \leq deg \varphi_i \leq d$ for every $1 \leq i \leq s$, we prove that \[ |A^{(s)}| +…

For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. In this paper, for a sequence of integers $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there exists an…

Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A, n)$ denote the number of solutions of the equation $n=a+b$ with $a, b\in A, a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal…

Let $A\subset [1,x]$ be a non-empty set of primes with $|A|= \alpha x(\log x)^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $|A+A|\geq c_1(\log x)(\log \log…

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors $a,b\in\mathbb{R}_+^n$ so that…

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, $E(A,A).$

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is…