Obstructions to Uniformity and Arithmetic Patterns in the Primes

2006/01/01 by Terence Tao · 2 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Mathematics #Arithmetic #Mathematics education

paper · pdf · doi:10.4310/pamq.2006.v2.n2.a2

openalex publication_date 2006/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/21

Abstract

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving "randomly", and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.

Citations

• Higher correlations of divisor sums related to primes III: k-correlations
• Regularity Lemma for k‐uniform hypergraphs
• The Gaussian primes contain arbitrarily shaped constellations
• On Some Sequences of Integers
• A variant of the hypergraph removal lemma
• Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes
• The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view
• The primes contain arbitrarily long arithmetic progressions
• Small gaps between primes
• On Certain Sets of Integers
• A quantitative ergodic theory proof of Szemerédi's theorem
• An inverse theorem for the Gowers U³ norm
• Roth's theorem in the primes

Cited by

• New bounds for Szemeredi's theorem, II: A new bound for r₄(N)
• Green–Tao theorem [wikipedia]