Multiple ergodic averages — Problem 30

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb P$ be the positive prime numbers, $\ell\in\mathbb N$, and $c,c_1,\ldots,c_\ell>0$ be arbitrary real numbers. Floors $\lfloor t\rfloor$ are integer parts; the exponents need not be distinct or nonintegral.

Statement recorded in the supplied source.

Do the primes contain both types of patterns \[\{m,m+\lfloor n^c\rfloor,m+2\lfloor n^c\rfloor,\ldots,m+\ell\lfloor n^c\rfloor\},\qquad \{m,m+\lfloor n^{c_1}\rfloor,\ldots,m+\lfloor n^{c_\ell}\rfloor\}?\] Each type is required to occur for infinitely many $n\in\mathbb N$, with a suitable starting value $m\in\mathbb N$ allowed to depend on $n$ and the type. [S2]

Short English statement

Do primes contain arbitrarily long progressions with power-floor differences and all specified mixed-power-floor patterns, for infinitely many input integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800030 (AMR-047-0030). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Section 6.3, full Problem 30, p.34. https://arxiv.org/pdf/1103.3808v3.