Approximate arithmetic structure in large sets of integers

2019/05/13 by Jonathan M. Fraser, Yu Han, Fraser, Jonathan M. +1
Computer Science · Mathematics · #11B05 #11B25 #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Numerical Methods and Algorithms

paper · pdf · doi:10.48550/arxiv.1905.05034

openalex publication_date 2019/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if a set is `large' in the sense of Erdős, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap length Δ of the progression, we improve a previous result of o(Δ) to O(Δ^α) for any α∈ (0,1).

Related

• Short Note: Every Large Set of Integers Contains a Three Term Arithmetic Progression
• Large subsets of Euclidean space avoiding infinite arithmetic progressions
• Sets with arithmetic progressions are abundant
• Improved bounds for arithmetic progressions in product sets
• Approximate algebraic structure