Large subsets of  without arithmetic progressions

Area of Science:

• Combinatorial Number Theory
• Set Theory
• Harmonic Analysis

Background:

• The study of arithmetic progressions is a fundamental problem in number theory.
• Finding maximal sets free from arithmetic progressions (AP) is a challenging task.
• Existing bounds for the size of such sets are often difficult to improve.

Purpose of the Study:

• To establish improved lower bounds for the size of progression-free sets in integers.
• To construct explicit examples of large progression-free sets.
• To advance the understanding of Behrend construction and its variations.

Main Methods:

• Construction of explicit progression-free sets using number-theoretic techniques.
• Analysis of the properties of these constructed sets concerning arithmetic progressions.
• Application of techniques from harmonic analysis and additive combinatorics.

Main Results:

• Improved lower bounds for the size of k-term progression-free sets in {1, ..., N}.
• Specific bounds derived for odd and even values of m, related to the least prime factor.
• Demonstration of new explicit constructions that outperform previous results for certain parameters.

Conclusions:

• The constructed sets provide significant improvements on known lower bounds for progression-free sets.
• The results contribute to the ongoing effort to determine the precise behavior of the maximum size of such sets.
• Further research directions include exploring these constructions for different settings and refining the bounds.