Enumeration of three term arithmetic progressions in fixed density sets

Abstract

Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemerédi's theorem using methods from real algebraic geometry.

62 pages. Update v2: Corrected some references. Update v3: Incorporated feedback

References in corpus (3)

• Symmetry groups, semidefinite programs, and sums of squares
• Some New Exact van der Waerden Numbers
• On the asymptotic minimum number of monochromatic 3-term arithmetic progressions