Related papers: Multi-fold sums from a set with few products

We prove that there exists a universal constant $c$ such that any finite primitive permutation group of degree $n$ with a non-trivial point stabilizer is a product of no more than $c\log n$ point stabilizers.

Let $\omega^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent…

Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$ such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k:…

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(\delta,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at…

Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

We prove that for any positive integer c there are at least N(c), $1\leq N(c) < \phi(c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc,…

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this…

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set…

Let G be a finite Abelian group and A be a subset G\times G of cardinality at least |G|^2/(log log |G|)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d does not equal 0. This…

The following theorem is proved. {\bf Theorem.} {\it Let $P(x) = \sum_{k=0}^{2n} a_k x^k$ be a polynomial with positive coefficients. If the inequalities $\frac{a_{2k+1}^2}{a_{2k}a_{2k+ 2}} < \frac{1}{cos^2(\frac{\pi}{n+2})} $ hold for all…

The Hardy--Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers $m,n\geq2$ and all $m$-linear forms $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$…

Let $A$ be a subset of integers and let $2\cdot A+k\cdot A=\{2a_1+ka_2 : a_1,a_2\in A\}$. Y. O. Hamidoune and J. Ru\' e proved that if $k$ is an odd prime and $A$ a finite set of integers such that $|A|>8k^k$, then $|2\cdot A+k\cdot A|\ge…

Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}}…

For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{\phi(q)} \sum_{\substack{\chi \text{ mod }q\\ \chi\neq \chi_0}} \Big| \sum_{n\leq x}…

We prove that, for every rational $d\ne 0,\pm 1$ and every compact set $K\subset\{s\in\mathbb{C}:1/2<\Re(s)<1\}$ with connected complement, any analytic non-vanishing functions $f_1,f_2$ on $K$ can be approximated, uniformly on $K$, by the…

Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k,…

We give an exact coefficients formula of any infinite product of power series with constant term equal to $1$, by using structures from partitions of integers and permutation groups. This is an universal theorem for various of Binomial-type…

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb…

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…