Quantitative concatenation for polynomial box norms
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 $1$.
Fields of Study & Subjects
Limits and Structures in Graph Theory
Analytic Number Theory Research
Advanced Topology and Set Theory
Publication details
- Journal / source
- Advances in Mathematics
- Volume
- 489
- Pages
- 110820–110820
- Publication date
- 28 January 2026
- Language
- EN