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POLYNOMIAL PATTERNS IN THE PRIMES
Forum of Mathematics, Pi ( IF 2.955 ) Pub Date : 2018-02-12 , DOI: 10.1017/fmp.2017.3
TERENCE TAO , TAMAR ZIEGLER

Let $P_{1},\ldots ,P_{k}:\mathbb{Z}\rightarrow \mathbb{Z}$ be polynomials of degree at most $d$ for some $d\geqslant 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are all not divisible by $p$. We show that there exist infinitely many natural numbers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_{1}(0)=\cdots =P_{k}(0)=0$ (though it allowed for the top degree coefficients to coincide). Furthermore, we obtain an asymptotic for the number of such prime pairs $n,m$ with $n\leqslant N$ and $m\leqslant M$ with $M$ slightly less than $N^{1/d}$. This asymptotic is already new in general in the homogeneous case $P_{1}(0)=\cdots =P_{k}(0)=0$. Our arguments rely on four ingredients. The first is a (slightly modified) generalized von Neumann theorem of the authors, reducing matters to controlling certain averaged local Gowers norms of (suitable normalizations of) the von Mangoldt function. The second is a more recent concatenation theorem of the authors, controlling these averaged local Gowers norms by global Gowers norms. The third ingredient is the work of Green and the authors on linear equations in primes, allowing one to compute these global Gowers norms for the normalized von Mangoldt functions. Finally, we use the Conlon–Fox–Zhao densification approach to the transference principle to combine the preceding three ingredients together. In the special case $P_{1}(0)=\cdots =P_{k}(0)=0$, our methods also give infinitely many $n,m$ with $n+P_{1}(m),\ldots ,n+P_{k}(m)$ in a specified set primes of positive relative density $\unicode[STIX]{x1D6FF}$, with $m$ bounded by $\log ^{L}n$ for some $L$ independent of the density $\unicode[STIX]{x1D6FF}$. This improves slightly on a result from our previous paper, in which $L$ was allowed to depend on $\unicode[STIX]{x1D6FF}$.

中文翻译:

质数中的多项式模式

$P_{1},\ldots ,P_{k}:\mathbb{Z}\rightarrow \mathbb{Z}$最多是次数多项式$d$对于一些$d\geqslant 1$, 与度$d$系数都是不同的,并且是可接受的,因为对于每个素数$p$, 存在整数$n,m$这样$n+P_{1}(m),\ldots ,n+P_{k}(m)$都不能被$p$. 我们证明存在无穷多个自然数$n,m$这样$n+P_{1}(m),\ldots ,n+P_{k}(m)$同时是素数,概括了作者先前的结果,该结果仅限于特殊情况$P_{1}(0)=\cdots =P_{k}(0)=0$(尽管它允许最高阶系数重合)。此外,我们获得了此类素数对数量的渐近线$n,m$$n\leqslant N$$m\leqslant M$$M$略低于$N^{1/d}$. 这种渐近线在齐次情况下一般来说已经是新的了$P_{1}(0)=\cdots =P_{k}(0)=0$. 我们的论点依赖于四个要素。第一个是作者的(略微修改的)广义冯诺依曼定理,将问题简化为控制冯曼戈尔特函数(适当的归一化)的某些平均局部高尔斯范数。第二个是作者最近的串联定理,通过全局 Gowers 规范控制这些平均的局部 Gowers 规范。第三个成分是 Green 和作者在素数线性方程组方面的工作,允许人们计算归一化 von Mangoldt 函数的这些全局 Gowers 范数。最后,我们对移情原理使用 Conlon-Fox-Zhao 致密化方法将前三种成分结合在一起。在特殊情况下$P_{1}(0)=\cdots =P_{k}(0)=0$,我们的方法也给出了无限多$n,m$$n+P_{1}(m),\ldots ,n+P_{k}(m)$在指定的正相对密度素数集合中$\unicode[STIX]{x1D6FF}$, 和$m$被约束$\log ^{L}n$对于一些$L$与密度无关$\unicode[STIX]{x1D6FF}$. 这比我们之前论文的结果略有改善,其中$L$被允许依赖$\unicode[STIX]{x1D6FF}$.
更新日期:2018-02-12
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