Linear Forms in Finite Sets of Integers

Let A_1, ..., A_r be finite, nonempty sets of integers, and let h_1,..., h_r be positive integers. The linear formh_1A_1 + ··· + h_rA_r is the set of all integers of the form b_1 + ··· + b_r, where b_i is an integer that can be represented as the sum ofh_i elements of the set A_i. In this paper, the structure of the linear form h_1A_1 + ··· + h_rA_r is completely determined for all sufficiently large integersh_i .     

对含有整数最小素因子的倒数的某些和的估计

设p(n)表示整数n>1的最小素因子.本文得到了(?),v≥0的渐近式,此处f_v(n)是一类数论函数.     

关于不同因子分解的数目

设f(n)表示分解自然数n(>1)为大于1的整数因子乘积的所有方式的数目(不计因子的顺序),并设0<β<1,N(x,β)=Card{n≤x,f(n)≥n~β}.本文分别估计了N(x,β)和f(n))的值.     

The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

A set A of non‐negative integers is called a Sidon set if all the sums , with and a₁, , are distinct. A well‐known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erd?s, Singer and Turán from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non‐differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erd?s (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015     

Dynamics of a Function Related to the Primes

Let n = p1p2 ··· pk, where pi(1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ωk(n) = P(p1 + p2)P(p2 + p3) ··· P(pk-1+ pk)P(pk+ p1), where P(n) is the largest prime factor of n. Define w0(n) = n and wi(n) = w(wi-1(n)) for all integers i ≥ 1. The smallest integer s for which there exists a positive integer t such thatΩs k(n) = Ωs+t k(n) is called the index of periodicity of n. The authors investigate the index of periodicity of n.     

Local resilience of graphs

In this article, we initiate a systematic study of graph resilience. The (local) resilience of a graph G with respect to a property measures how much one has to change G (locally) to destroy . Estimating the resilience leads to many new and challenging problems. Here we focus on random and pseudorandom graphs and prove several sharp results. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

数论函数 $\Omega_k$ 的动力系统

本文推广了Goldring$^{[1]}$关于一个数论函数的动力系统的相关结果.     

The Distribution of {alpha}p Modulo One

In this paper the authors prove the following result. Let be an irrational number. Then for any > 0, there areinfinitely many prime numbers p such that | p | < p^–16/49+. The exponent improves on , which was obtained recently by thesecond author [Sci. China Ser. A 43 (2000) 703-721]. The resultis very close to the exponent , which can be obtained under the Generalized Riemann Hypothesis. Previous approaches to this problem have all used the same basicestimates for the trigonometric sums that arise. However, thepresent proof uses new bounds, which depend on the Kloostermansum and also on a counting problem in the geometry of numbers.In addition new techniques for the sieve method are applied.The most significant feature of the new approach is that, unlikeprevious methods, the exponential sum estimates remain non-trivialfor the exponent . This gives one hope for an unconditional result as good as that availableunder the Generalized Riemann Hypothesis. 2000 MathematicalSubject Classification: 11N36, 11J71.     

Ratios of Norms for Polynomials and Connected n-width Problems

Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:

On sums and products of integers

Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

     

On sums and products of integers

Erdos and Szemerédi conjectured that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of . Erdos and Szemerédi proved that this number must be at least for some and . In this paper it is proved that the result holds for .

     

On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs.Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques.This connection provided a new proof of Turán classical result on the Turán density of complete graphs.Since then,Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs.Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range.They showed that if G is a 3-uniform graph with m edges containing a clique of order t-1,then λ(G)=λ([t-1]~((3))) provided (t-13)≤m≤(t-13)+_(t-22).They also conjectured:If G is an r-uniform graph with m edges not containing a clique of order t-1,then λ(G)λ([t-1]~((r))) provided (t-1r)≤ m ≤(t-1r)+(t-2r-1).It has been shown that to verify this conjecture for 3-uniform graphs,it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m=t-13+t-22.Regarding this conjecture,we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t-1]~((3))\E(G)|=p,then λ(G)λ([t-1]~((3))) provided m=(t-13)+(t-22) and t≥17p/2+11.     

On the distribution of

Suppose that α is an irrational number and β is a real number. It is proved that there are infinitely many prime numbers p such that ‖ αp-β‖ <p ^-9/28.     

一个素数不等式的加强

对于一个有关素数的级数,通过构造不等式得到了级数和的更小的上界.     

Unique factorization in cyclotomic integers of degree seven

This article provides a survey of some basic results in algebraic number theory and applies this material to prove that the cyclotomic integers generated by a seventh root of unity are a unique factorization domain. Part of the proof uses the computer algebra system Maple to find and verify factorizations. The proofs use a combination of historic and modern techniques and some attempt has been made to discuss the history of this material.     

Random coloring method in the combinatorial problem of Erdős and Lovász

The work deals with a combinatorial problem of P. Erd?s and L. Lovász concerning simple hypergraphs. Let denote the minimum number of edges in an n‐uniform simple hypergraph with chromatic number at least . The main result of the work is a new asymptotic lower bound for . We prove that for large n and r satisfying the following inequality holds where . This bound improves previously known bounds for . The proof is based on a method of random coloring. We have also obtained results concerning colorings of h‐simple hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

On the maximum number of copies of H in graphs with given size and order

We study the maximum number $ex\left(n,e,H\right)$ of copies of a graph $H$ in graphs with a given number of vertices and edges. We show that for any fixed graph $H,ex\left(n,e,H\right)$ is asymptotically realized by the quasi‐clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.     

Sign changes of π (x,q,1) - π(x,q,a)

It is known that under the assumption of the generalized Riemann hypothesis the function π(x,q,1) - π(x,q,a) has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly, assuming the Riemann hypothesis we give a lower bound for the number of sign changes of π(x)-li x. The implied results for the least sign change are weaker than those obtained by numerical methods, however, our method makes no use of computations of zeros of the ζ-function. This revised version was published online in June 2006 with corrections to the Cover Date.     

The largest prime factor of X3+2

The largest prime factor of X³+2 was investigated in 1978 byHooley, who gave a conditional proo that it is infinitely oftenat least as large as X¹+, with a certain positive constant .It is trivial to obtain such a result with =0. One may thinkof Hooley's result as an approximation to the conjecture thatX³+2 is infinitely often prime. The condition required by Hooley,his R^* conjecture, gives a non-trivial bound for short Ramanujan–Kloostermansums. The present paper gives an unconditional proof that thelargest prime factor of X³+2 is infinitely often at least aslarge as X¹⁺, though with a much smaller constant than thatobtained by Hooley. In order to do this we prove a non-trivialbound for short Ramanujan–Kloosterman sums with smoothmodulus. It is also necessary to modify the Chebychev method,as used by Hooley, so as to ensure that the sums that occurdo indeed have a sufficiently smooth modulus. 2000 MathematicsSubject Classification: 11N32.