Uniformity seminorms on ℓ∞ and applications

@article{Host2009UniformitySO,
title={Uniformity seminorms on ℓ∞ and applications},
author={Bernard Host and Bryna Kra},
journal={Journal d'Analyse Math{\'e}matique},
year={2009},
volume={108},
pages={219-276}
}
• Published 11 September 2009
• Mathematics
• Journal d'Analyse Mathématique
A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer k ≥ 1…
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