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Polynomial configurations in dense subsets of the prime lattice

Andrew Lott, Ákos Magyar, Giorgis Petridis, János Pintz

Abstract

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form $x+P_0(y)v_0,\ldots, x+P_l(y)v_l$, for some $x$ in $\mathbb{Z}^d$ and $y$ in $\mathbb{N}$, which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if $A$ is a subset of the first $N$ positive integers.

Polynomial configurations in dense subsets of the prime lattice

Abstract

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let be a subset of the prime lattice - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form , for some in and in , which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if is a subset of the first positive integers.

Paper Structure

This paper contains 5 sections, 12 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. The pseudo-random majorant and the polynomial forms condition.
  3. PET induction and the generalized von Neumann inequality.
  4. Dual functions and the main decomposition.
  5. Proof of Theorem 1.

Key Result

Proposition 2.1

Let $t,d,D,J$ be fixed natural numbers, let $\varepsilon_0$ sufficiently small and $L$ sufficiently large depending on $t,d,D,I$. Let $\underline{\mathcal{Q}}=(\underline{Q}_1,\ldots,\underline{Q}_J):\mathbb{Z}^t\to\mathbb{Z}^d$ be a non-degenerate integral polynomial map of degree at most $D$, with where $H=\log^{\sqrt{L}} N$ and $X=(\mathbb{Z}/N\mathbb{Z})^d\simeq [N]^d$.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • proof : Proof of Proposition \ref{['prop1']}
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 17 more