Expanding total sieve and patterns in primes

Abstract

Let $({\mathcal{S}}_n^{\alpha ,\kappa ,\mathfrak{r}}(z){)}_{n=1}^{\infty }$ be a sequence of the largest possible integer intervals, such that $z\in {\mathcal{S}}_n^{\alpha ,\kappa ,\mathfrak{r}}(z)\subset {\overline{\mathcal{M}}}_n^{\alpha ,\kappa ,\mathfrak{r}}={\bigcup }_{i=1}^n[{\mathfrak{r}}_i{]}_{{\mathfrak{p}}_i}$ or ${\mathcal{S}}_n^{\alpha ,\kappa ,\mathfrak{r}}(z)=\varnothing$, where ${\mathfrak{p}}_i=p_{\alpha +\lceil i/\kappa \rceil -1}$ and $z\in \mathbb{Z}$. We prove that $(\#{\mathcal{S}}_n^{\alpha ,\kappa ,\mathfrak{r}}(z){)}_{n=1}^{\infty }$ oscillates infinitely many times around ${\beta }_n=o\left(n^2\right)$ for any fixed $\alpha \in {\mathbb{Z}}^{+}$, $\kappa \in \mathbb{Z}\cap [1,p_{\alpha })$, and ${\mathfrak{r}}_i\in \mathbb{Z}$. Let $T=(a_1,a_2,\dots ,a_k)$ be an admissible $k$-tuple and let ${\mathcal{X}}_n^{T,k,\rho ,\eta }=\left\{x\in [\rho {]}_{\eta }:\{x+a_1,x+a_2,\dots ,x+a_k\}\cap {\mathcal{M}}_{n+\alpha -1}\ne \varnothing \right\}$ for each $n\in {\mathbb{Z}}^{+}$, where ${\mathcal{M}}_g={\bigcup }_{i=1}^g[0{]}_{p_i}$. We prove that for any $T$ and for some fixed $\alpha$, $\kappa$, $\rho$, $\eta$, $z$, and $\mathfrak{r}$, there exists a linear bijection between ${\overline{\mathcal{M}}}_{\kappa n}^{\alpha ,\kappa ,\mathfrak{r}}$ and ${\mathcal{X}}_n^{T,k,\rho ,\eta }$ for each $n\in {\mathbb{Z}}^{+}$. It implies that the length of any expanding integer interval on which all occurrences of $T$ are sieved out by ${\mathcal{M}}_{n+\alpha -1}$ oscillates infinitely many times around ${\tilde{\beta }}_n=o\left(n^2\right)$. The concept of the sieve of Eratosthenes asserts ${\mathcal{E}}_n=[2,p_{n+\alpha }^2)\cap \left(\mathbb{Z}\setminus {\mathcal{M}}_{n+\alpha -1}\right)\subset \mathbb{P}$. Therefore, having $p_{n+\alpha }^2=\omega \left(n^2\right)$, we obtain that ${\mathcal{E}}_n$ includes a subset matched to $T$ for infinitely many values of $n$ and, consequently, $T$ matches infinitely many positions in the sequence of primes.

14 pages, 4 figures, comments and constructive criticism are welcome, especially regarding legibility, correctness and completeness

References in corpus (1)

• The "bounded gaps between primes" Polymath project - a retrospective