A Density Increment Approach to Roth's Theorem in the Primes

2014/09/11 by Eric Naslund, Naslund, Eric
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1409.3595

openalex publication_date 2014/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if A is any set of prime numbers satisfying ∑_{a∈ A}(1)/(a)=∞, then A must contain a 3-term arithmetic progression. This is accomplished by combining the transference principle with a density increment argument, exploiting the structure of the primes to obtain a large density increase at each step of the iteration. The argument shows that for any B>0, and N>N₀(B), if A is a subset of primes contained in \1,…,N\ with relative density α(N)=(|A|log N)/N at least α(N)≫_B(loglog N)^-B then A contains a 3-term arithmetic progression.

Citations

• Roth’s theorem on progressions revisited
• On Certain Sets of Integers

Related

• On arithmetic progressions in A + B + C
• Roth's theorem in the primes
• The primes contain arbitrarily long arithmetic progressions
• A new proof of Roth's theorem on arithmetic progressions
• Improving Roth's theorem in the primes