Caps and progression-free sets in

Area of Science:

• Combinatorial Number Theory
• Additive Combinatorics
• Abstract Algebra

Background:

• Progression-free sets are fundamental in additive combinatorics.
• Understanding the maximum size of such sets in abelian groups is a key challenge.
• Existing bounds for affine caps, sets without 3 collinear points, require improvement.

Purpose of the Study:

• To establish new lower bounds for the size of progression-free sets in abelian groups.
• To develop novel methods for constructing these sets.
• To improve asymptotic lower bounds for affine caps.

Main Methods:

• Construction of lower bound examples for progression-free sets.
• Development of new techniques applicable to various parameter families.
• Analysis of specific cases for exact value determination.

Main Results:

• New lower bounds for progression-free sets in $$ groups, including $$ for even $$.
• Order of magnitude lower bound of $$ for $$.
• Exact values determined for $$ when $$ and for $$ when $$.
• Asymptotic improvements for affine caps in $$ and $$.

Conclusions:

• The study provides significant advancements in the understanding of progression-free sets.
• New methods offer substantial improvements, particularly for affine caps.
• The results open avenues for further research in combinatorial number theory.