相关论文: The sum-product problem for integers with few prim…

We prove that every additive set $A$ with energy $E(A)\ge |A|^3/K$ has a subset $A'\subseteq A$ of size $|A'|\ge (1-\varepsilon)K^{-1/2}|A|$ such that $|A'-A'|\le O_\varepsilon(K^{4}|A'|)$. This is, essentially, the largest structured set…

We prove that there are infinitely many integers $n$ such that the total number of prime factors of $(n+h_{1})(n+h_{2})...(n+h_{\kappa})$ is at most $(1/2)\kappa\log\kappa+O(\kappa)$, provided $\kappa$ is sufficiently large.

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii)…

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s} q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1, \ldots, \beta_t$ bounded by a given constant.…

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also…

We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j,…

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound…

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)}| +…

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…

Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the…

The summation formula $$ \sum^{n-1}_{i=0}\epsilon^i i! (i^k+u_k) = v_k+\epsilon^{n-1} n! A_{k-1}(n) $$ $(\epsilon=\pm 1; k=1,2,...; u_k, v_k\in \msbm\hbox{Z}; A_{k-1}$ is a polynomial) is derived and its various aspects are considered. In…

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szab\'{o} Theorem…

Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there…

For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $\lambda \in \mathbb{Z}$, the sum of dilates $A + \lambda \cdot A$ is defined by \[A + \lambda \cdot A = \{a + \lambda a' : a, a' \in A\}.\] The basic problem on such sums of dilates…

Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently…

Professor Georges Rhin considers a nonzero algebraic integer $\a$ with conjugates $\a_1=\a, \ldots, \a_d$ and asks what can be said about $\d \sum_{ | \a_i | >1} | \a_i |$, that we denote ${\rm{R}}(\a)$. If $\a$ is supposed to be a totally…

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement…

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of…

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d.…

We consider the problem of finding the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized. While the set of permutations maximizing this value are easily determined,…