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Quantitative twisted patterns in positive density subsets

Kamil Bulinski, Alexander Fish

Abstract

We make quantitative improvements to recently obtained results on the structure of the image of a large difference set under certain quadratic forms and other homogeneous polynomials. Previous proofs used deep results of Benoist-Quint on random walks in certain subgroups of $\operatorname{SL}_r(\mathbb{Z})$ (the symmetry groups of these quadratic forms) that were not of a quantitative nature. Our new observation relies on noticing that rather than studying random walks, one can obtain more quantitative results by considering polynomial orbits of these group actions that are not contained in cosets of submodules of $\mathbb{Z}^r$ of small index. Our main new technical tool is a uniform Furstenberg-Sárközy theorem that holds for a large class of polynomials not necessarily vanishing at zero, which may be of independent interest and is derived from a density increment argument and Hua's bound on polynomial exponential sums.

Quantitative twisted patterns in positive density subsets

Abstract

We make quantitative improvements to recently obtained results on the structure of the image of a large difference set under certain quadratic forms and other homogeneous polynomials. Previous proofs used deep results of Benoist-Quint on random walks in certain subgroups of (the symmetry groups of these quadratic forms) that were not of a quantitative nature. Our new observation relies on noticing that rather than studying random walks, one can obtain more quantitative results by considering polynomial orbits of these group actions that are not contained in cosets of submodules of of small index. Our main new technical tool is a uniform Furstenberg-Sárközy theorem that holds for a large class of polynomials not necessarily vanishing at zero, which may be of independent interest and is derived from a density increment argument and Hua's bound on polynomial exponential sums.

Paper Structure

This paper contains 11 sections, 23 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Recurrent orbits
  3. A uniform Furstenberg-Sárközy theorem
  4. Polynomial exponential bounds
  5. $(q,\delta)$-Equidistributed sets.
  6. A quantitative polynomial mean ergodic theorem and proof of Theorem \ref{['thm: uniform polynomial recurrence']}
  7. Constructing polynomials from unipotent elements (proof of Theorem \ref{['thm: uniformly virtually recurrent orbits for unipotents']})
  8. Applications to the adjoint representation and the proof of Theorem \ref{['thm: uniform virtual recurrence for adjoint representation']}
  9. Quadratic form $x^2 - y^2 - z^2$ and the completion of the proof of Theorem \ref{['thm: example of uniform virtually recurrent level sets']}
  10. Quantitative polynomial Bogolyubov's theorem
  11. Some algebraic facts

Key Result

Theorem 1.1

For all $\epsilon>0$ and integers $d \geq 5$ there exists a positive integer $k=k(\epsilon,d)>0$ such that the following holds: If $B \subset \mathbb{Z}^d$ has upper Banach density then there exists a positive integer $N_0=N_0(B)$ such that

Theorems & Definitions (49)

  • Theorem 1.1: Magyar Magyar
  • Theorem 1.2: BFChar
  • Theorem 1.3
  • Theorem 1.4: Polynomial Bogolyubov's theorem
  • Definition 1.5
  • Proposition 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10: BFChar
  • ...and 39 more