[Paper Review] The polynomial multidimensional Szemerédi Theorem along shifted primes

Nikos Frantzikinakis, Bernard Host+1 more|arXiv (Cornell University)|Sep 8, 2010

Analytic Number Theory Research17 references4 citations

TL;DR

This paper establishes the polynomial multidimensional Szemerédi theorem along shifted primes, proving that any subset of ℤ^ℓ with positive upper Banach density contains configurations defined by polynomial shifts evaluated at shifted primes. The proof combines uniformity norms from higher-order Fourier analysis with ergodic theory and prime number theory, extending prior results on arithmetic progressions and recurrence in sparse sets.

ABSTRACT

If $\vf_1, ... \vf_m\colon\Z o\Z^\ell$ are polynomials with zero constant terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for some prime $p$. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.

Motivation & Objective

• To extend the polynomial multidimensional Szemerédi theorem to configurations defined by shifted primes, generalizing earlier results on arithmetic progressions in dense sets.
• To establish that for any set of positive upper Banach density in ℤ^ℓ, there exist shifts by p−1 (p prime) that preserve intersection patterns defined by polynomial mappings.
• To prove mean convergence of multiple ergodic averages along primes, conditional on convergence along the full integers, using weighted averages with von Mangoldt-type weights.
• To unify and generalize prior results by Wooley–Ziegler, Bergelson–Leibman–Ziegler, and the authors’ earlier work on linear and quadratic patterns in primes.
• To demonstrate that the set of parameters n for which recurrence occurs has positive relative density within the shifted primes ℙ−1 and ℙ+1.

Proposed method

• Uses the Furstenberg Correspondence Principle to translate the combinatorial problem into an ergodic-theoretic recurrence problem on a probability space with commuting measure-preserving transformations.
• Applies a uniform version of the polynomial Szemerédi theorem (Theorem 4.1 and Corollary 4.2) to ensure uniform lower bounds on multiple recurrence independent of the specific transformations.
• Employs the von Mangoldt-type weight function Λ′(n) derived from the W-trick and nilsequence equidistribution to model the distribution of shifted primes in arithmetic progressions.
• Relies on the prime number theorem in arithmetic progressions and the linear forms condition to control the uniformity of the weight sequence Λ′(n) over intervals.
• Uses the method of majorizing measures and majorizing sequences to reduce convergence of averages along primes to convergence along arithmetic progressions.
• Applies the Cauchy criterion in L²(μ) to prove convergence of weighted multiple ergodic averages, leveraging the relative density of shifted primes and the structure of nilsequences.

Experimental results

Research questions

• RQ1 Can the polynomial multidimensional Szemerédi theorem be restricted to parameters along shifted primes, i.e., p−1 and p+1 for p prime?
• RQ2 Does the set of integers n for which multiple recurrence occurs along polynomial shifts have positive relative density in the shifted primes?
• RQ3 Under what conditions do multiple ergodic averages along primes converge in L²(μ), given convergence along the full integers?
• RQ4 Can the uniform multiple recurrence result be adapted to sequences supported on shifted primes via the W-trick and nilsequence theory?
• RQ5 How does the structure of the von Mangoldt function and its majorized version relate to the equidistribution of polynomial orbits in nilsystems?

Key findings

• For any set E ⊂ ℤ^ℓ with positive upper Banach density and any collection of polynomial mappings q_i: ℤ → ℤ^ℓ with zero constant term, the intersection E ∩ (E − q₁(p−1)) ∩ … ∩ (E − q_m(p−1)) is nonempty for some prime p.
• The set of primes p for which such recurrence occurs has positive relative density within the shifted primes ℙ−1 and ℙ+1.
• Mean convergence of multiple ergodic averages along primes holds in L²(μ) whenever the corresponding averages along ℕ converge, conditional on the convergence assumption.
• The convergence of the weighted averages ∑_{p≤N} f₁(T^{q₁(n)}x)⋯f_m(T^{q_m(n)}x) / π(N) is established via the Cauchy criterion, using majorizing sequences and uniformity of the weight function.
• The proof establishes that the limit inferior of the multiple recurrence average along Wn+r (for fixed W and r coprime to W) is bounded below by a positive constant depending only on the measure of the set and the degree of the polynomials.
• The result generalizes previous theorems by Wooley–Ziegler (for ℓ=1) and Bergelson–Leibman–Ziegler (for linear polynomials), extending them to arbitrary polynomial mappings with zero constant terms.