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First page of Polynomial configurations and pointwise averages along Piatetski–Shapiro primes

Polynomial configurations and pointwise averages along Piatetski–Shapiro primes

Hamed Mousavi

math.CO Nov 3, 2026 · v1 math.DS math.NT
formalization number-theory combinatorics
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TL;DR
An accompanying, publicly available Lean development formalizes parts of the paper's arguments; external cited results and interface translations are left unproved in Lean.
Abstract
For every \(c>1\) sufficiently close to \(1\), we prove that there is \(κ>0\) such that every sufficiently large \(A\subset[N]\) of density at least \((\log N)^{-κ}\) contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.
Problem

The paper asks whether dense subsets of [N], with density at least (log N)^{-κ}, must contain polynomial configurations x, x+P_1(p-1), ..., x+P_ℓ(p-1) when p ranges over Piatetski–Shapiro primes ⌊n^c⌋, which have zero relative density among the primes. It also asks whether the corresponding multiple ergodic averages converge pointwise.

Approach

A transference argument compares Piatetski–Shapiro prime weights with ordinary prime weights. It uses a polynomial-phase estimate derived from the Shao–Sun nilsequence theorem. That comparison is inserted into the distinct-degree density increment argument of Krause–Mousavi–Tao–Teräväinen, and a pointwise comparison along lacunary subsequences handles the ergodic averages. A Lean development accompanying the paper, generated with ChatGPT assistance, formalizes parts of the argument.

Results

For c>1 sufficiently close to 1, sets of density at least (log N)^{-κ} contain the configurations, and the multiple ergodic averages converge pointwise almost everywhere. The Lean development does not prove the external results it relies on and has not yet been independently reviewed by human experts.