Short Note: Every Large Set of Integers Contains a Three Term Arithmetic Progression

2014/04/06 by Korvin, Gabor
#11B25 Arithmetic progressions #11B75 Other combinatorial number theory #11Pxx Additive number theory #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1404.1557

Abstract

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum of reciprocals of its elements diverges then the sequence contains an arithmetic progression of length three.

Related

• Strong Bounds for 3-Progressions
• Four-term progression free sets with three-term progressions in all\n large subsets
• Long Arithmetic Progressions in Critical Sets
• Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions
• Roth's theorem in the primes