A Multidimensional Szemerédi Theorem in the Primes via Combinatorics

Let A be a subset of positive relative upper density of P d , the d -tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set F ⊆ Z d . This provides a natural multidimensional extension of the theorem...

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Bibliographic Details
Published in:Annals of combinatorics 2018, Vol.22 (4), p.711-768
Main Authors: Cook, Brian, Magyar, Ákos, Titichetrakun, Tatchai
Format: Article
Language:English
Subjects:
Combinatorial analysis
Combinatorics
Existence theorems
Mathematics
Mathematics and Statistics
Progressions
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Summary:Let A be a subset of positive relative upper density of P d , the d -tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set F ⊆ Z d . This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes.
ISSN:0218-0006
0219-3094
DOI:10.1007/s00026-018-0402-4