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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.