Result: Asymptotic Behavior of r sub (k(N ))(N ): Progression-Free Sets with Growing Length

Title:
Asymptotic Behavior of r sub (k(N ))(N ): Progression-Free Sets with Growing Length
Authors:
Fathi, Kevin
Publication Status:
Preprint
Publisher Information:
Zenodo, 2025.
Publication Year:
2025
Subject Terms:
Additive Combinatorics, Asymptotic Analysis, Combinatorial Number Theory, Elementary Methods, Arithmetic Progressions, Progression-Free Sets, Variable Progression Length, Behrend Construction, Clause Complexity, Szemerédi-Type Theorems, Density Threshold, Erdős Problems, Clause Overlap, Forbidden Configurations, Pigeonhole Principle, Entropy Saturation, Growth Barrier, Extremal Set Theory
Language:
English
DOI:
10.5281/zenodo.15611840
DOI:
10.5281/zenodo.15611839
Rights:
CC BY
Accession Number:
edsair.doi.dedup.....576b64b2cc945e7b489d3cfe64d30c9f
Database:
OpenAIRE

Further Information

This paper determines the asymptotic behavior of the function r_k(N), which represents the maximum size of a subset of the integers from 1 to N that contains no nontrivial arithmetic progression of length k. Unlike previous results that focused on fixed values of k, we analyze the case where k grows slowly with N. Using a combinatorial framework based on clause saturation and density arguments, we prove that if k exceeds a constant times log N divided by log log N, then any subset of positive density must contain a k-term arithmetic progression. This resolves a major open question originally posed by Erdős and extends classical results by Roth and Szemerédi. The argument avoids analytical or Fourier-based methods and is entirely elementary and combinatorial in nature.