Multidimensional Schinzel-type theorems for multiplicative functions near prime tuples

3-tuples have at most 7 prime factors infinitely often

Let $L_1$, $L_2$ $L_3$ be integer linear functions with no fixed prime divisor. We show there are infinitely many $n$ for which the product $L_1(n)L_2(n)L_3(n)$ has at most 7 prime factors, improving a result of Porter. We do this by means of a weighted sieve based upon the Diamond-Halberstam-Richert multidimensional sieve.

A short proof of the multidimensional Szemerédi theorem in the primes

Tao conjectured that every dense subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This was very recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler. Here we give a simple proof using the Green-Tao theorem on linear equations in primes and the Furstenberg-Katznelson multidimensional Szemerédi theorem.

素数中的多维 Szemerédi 定理

A Multidimensional Szemerédi Theorem in the primes

本文证明素数多维组中任意有限集均存在仿射副本，推广了Green-Tao定理，其通过非均匀超图伪随机权重系统并扩展超图移除引理证明完成。

整数中的多维Szemerédi定理

A multidimensional Szemerédi theorem in integers

该研究证明了在整数n维空间中，密度足够大的子集包含特定配置，扩展了Peluse等人关于二维配置的结果。