The Monks-Peluse-Ye theorem on consecutive primes in arithmetic progressions

It has already been noted on this blog that REUs seem to be great opportunities for undergrads (and sometimes high schoolers) to produce interesting research.

The one that happened at Emory this last summer is a new impressive example: three instructors (among whom the famous Ken Ono) tutored nine talended undergrads, and within the six weeks of the program they produced six interesting papers!

The one that actually got me to that webpage is by three of the undergradutes: Monks, Peluse and Ye studied the interactions between the Green-Tao theorem on primes in arithmetic progressions and Shiu’s results on consecutive primes within a same congruence class. They obtained:

Theorem (Monks-Peluse-Ye) If the set of primes \mathcal A is well-distributed (a technical definition) then there exist arbitrarily long strings of consecutive primes in \mathcal A all of which lie in any given progression \{a + qn\} with \gcd(q, a) = 1.

They actually have a quantitative version which bounds from below the length of the arithmetic progression by (a function of) the the size of \mathcal A.  And as a corollary they have:

Corollary (Monks-Perulse-Ye) For any irrational \alpha > 0 of fi nite type (i.e. most irrationals), there exist arbitrarily long strings of consecutive primes in {\mathcal B}_{\alpha}=\{ { \lfloor \alpha \rfloor } , {\lfloor 2\alpha \rfloor } , {\lfloor 3\alpha \rfloor } , \cdots \} all of which lie in any progression \{a+qn\} with \gcd(q, a) = 1.

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