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Quantitative concatenation for polynomial box norms

Noah Kravitz, Borys Kuca, James Leng

TL;DR

This work develops a quantitative concatenation framework for multidimensional polynomial progressions by combining PET induction with a two-stage approach: duplicating variables via Cauchy–Schwarz and establishing equidistribution for multilinear systems, all in the integer setting. The main achievement is a box-norm control result: if a counting operator for a polynomial progression is large, then, for each index, one obtains a lower bound on a high-degree box norm along a carefully chosen set of directions determined by the leading coefficients and differences of the polynomials. The authors introduce a robust notion of normal polynomial families and a van der Corput-based descent that preserves essential structural properties, enabling iterative concatenation of averages of box norms into a single controlling box norm with polynomial losses. These results extend prior finite-field and one-dimensional methods to complex multidimensional integer configurations, with applications to quantitative bounds for sets avoiding polynomial progressions and broader implications in ergodic theory and harmonic analysis. The work lays a foundation for explicit bounds in higher-dimensional polynomial patterns and has already inspired downstream results in adjacent areas.

Abstract

Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper arXiv:2407.08637, we complete this program for sets in $[N]^2$ lacking nondegenerate progressions of the form $(x, y), (x + P(z), y), (x, y + P(z))$, where $P \in \mathbb{Z}[z]$ is any fixed polynomial with an integer root of multiplicity $1$.

Quantitative concatenation for polynomial box norms

TL;DR

This work develops a quantitative concatenation framework for multidimensional polynomial progressions by combining PET induction with a two-stage approach: duplicating variables via Cauchy–Schwarz and establishing equidistribution for multilinear systems, all in the integer setting. The main achievement is a box-norm control result: if a counting operator for a polynomial progression is large, then, for each index, one obtains a lower bound on a high-degree box norm along a carefully chosen set of directions determined by the leading coefficients and differences of the polynomials. The authors introduce a robust notion of normal polynomial families and a van der Corput-based descent that preserves essential structural properties, enabling iterative concatenation of averages of box norms into a single controlling box norm with polynomial losses. These results extend prior finite-field and one-dimensional methods to complex multidimensional integer configurations, with applications to quantitative bounds for sets avoiding polynomial progressions and broader implications in ergodic theory and harmonic analysis. The work lays a foundation for explicit bounds in higher-dimensional polynomial patterns and has already inspired downstream results in adjacent areas.

Abstract

Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper arXiv:2407.08637, we complete this program for sets in lacking nondegenerate progressions of the form , where is any fixed polynomial with an integer root of multiplicity .
Paper Structure (20 sections, 28 theorems, 341 equations, 1 figure)

This paper contains 20 sections, 28 theorems, 341 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Further applications
  3. Acknowledgments
  4. Proof strategy
  5. Comparison with previous approaches
  6. Definitions and preliminaries
  7. Basic notation
  8. Van der Corput inequalities
  9. Box norms
  10. PET induction scheme
  11. Example of PET
  12. Setting up the induction scheme
  13. Bounds by averages of box seminorms
  14. Model example: concatenation of degree-$1$ box norms
  15. Sketch of the proof of Proposition \ref{['P: concatenation 1-deg']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $d, \ell,D \in {\mathbb N}$ and $T>0$. There exist a positive integer $t=O_{d, \ell}(1)$ and a positive real $C=O_{d,\ell,D,T}(1)$ such that the following holds. Let $N \in {\mathbb N}$ and $\delta>0$ satisfy $N\geq C\delta^{-C}$, and let ${\mathbf{P}}_1, \ldots, {\mathbf{P}}_\ell\in{\mathbb Z}^ implies that for all $j\in[0,\ell]$, where $d_{jj'} = \deg ({\mathbf{P}}_j - {\mathbf{P}}_{j'})$ a

Figures (1)

  • Figure 1: The main steps and logical implications in the proof of Theorem \ref{['T: single box norm bound']}.

Theorems & Definitions (64)

  • Theorem 1.1
  • Example 1.2
  • Corollary 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 54 more