Generalized additive bases, Konig's lemma, and the Erdos-Turan conjecture

On a conjecture of ErdŐs, Pólya and Turán on consecutive gaps between primes

Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erdős and Turán showed that d_{n+1}-d_n changes sign infinitely often and together with Pólya asked for a necessary and sufficient condition that a fixed linear combination of r>1 consecutive prime gaps d_{n+i} i=(1,2,...r) should change sign infinitely often as n runs through the sequence of all natural numbers. They conjectured that the condition is that the non zero values among the coefficients of the linear combination cannot all have the same sign but they…

Erdos-Turan with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also…

关于 Erdős-Turán 猜想与 $B_{2}[g]$ 序列的增长

On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences

本文研究Erdős-Turán猜想中B₂[g]序列的加法基性质与增长密度；通过构造满足两两和表示数受限的序列，证明在g>1/ε时其可带小项三项表示所有充分大整数，并将计数下界改进为≫x^{g/(2g+1)}。

On some properties of representation functions related to the Erdős-Turán conjecture

For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erdős-Turán conjecture says that, if $R_A(n)\ge 1$ for all sufficiently large integers $n$, then the representation function $R_A(n)$ cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is defined to be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_m$ with $1\le R_A(n)\le r$ for all $n\in \mathbb{Z}_m$. In 2008, Chen proved that $R_{m}\le 288$ for all positive integers $m$. Recent…

On some power sum problems of Montgomery and Turan

We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h=>2 be an integer. We prove that inf_{|z_k| => 1} max_{v=1,...,n^h} |sum_{k=1}^n z_k^v| <= (h-1+o(1)) sqrt n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order sqrt log n.

Turan's problem 10 revisited

In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case of $n+1$ being a prime we have previously proved the much sharper result that the quantity lies in the interval [sqrt(n),sqrt(n+1)] The method of proof combines a general lower bound (of Andersson), explicit arithmetical constructions (of Montgomery, Fabrykowski or Andersson), moments (probabilistic methods) and estimates for the difference of consecutive primes (of Baker, Harman and Pintz). We al…

加性组合学中的罗思型定理

Roth-type theorems in additive combinatroics

本文探讨加性组合学中罗思型定理的发展，介绍了其起源、核心问题及主要研究成果。

关于埃尔多斯和拉马努金除数和的求和

On divisor sums due to Erdős and Ramanujan

本文研究了埃尔多斯和拉马努金除数和的混合形式，通过结合戈尔本的估计和塔兰的定量化形式，证明了除数和的渐进行为。