Residue Class Patterns of Consecutive Primes

For $m,q\in \mathbb{N}$, we call an $m$-tuple $(a_1,\dots ,a_m)\in {\prod }_{i=1}^m(\mathbb{Z}\mathrm{/}q\mathbb{Z}{)}^{\times }$ good if there are infinitely many consecutive primes $p_1,\dots ,p_m$ satisfying $p_i\equiv a_i(\mathrm{m}\mathrm{o}\mathrm{d}q)$ for all $i$. We show that given any $m$ sufficiently large, $q$ squarefree, and $A\subseteq (\mathbb{Z}\mathrm{/}q\mathbb{Z}{)}^{\times }$ with $\mathrm{\mid }A\mathrm{\mid }=\lfloor 71(\log m{)}^3\rfloor$, we can form at least one non-constant good $m$-tuple $(a_1,\dots ,a_m)\in {\prod }_{i=1}^{m}A$. Using this, we can provide a lower bound for the number of residue class patterns attainable by consecutive primes, and for $m$ large and $\phi (q)\gg (\log m{)}^{10}$ this improves on the lower bound obtained from direct applications of Shiu (2000) and Dirichlet (1837). The main method is modifying the Maynard-Tao sieve found in Banks, Freiberg, and Maynard (2015), where instead of considering the 2nd moment we considered the $r$-th moment, where $r$ is an integer depending on $m$.