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Quantitative concatenation for polynomial box norms (2407.08636v2)

Published 11 Jul 2024 in math.CO and math.NT

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 $1$.

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