等差數列集合密度的研究與進展

Abstract

Roth 定理是加性組合中的一個知名定理，此定理的敘述為：給定一個正整數子集合，且此集合不包含任何長度為 3 的等差數列，則我們能給予此集合大小的估計。在 Roth 定理之後，有許多數學家對此集合的大小估計進行改進，在此篇文章哩，我們將介紹 Bourgain 與 Sanders 對此集合大小估計的改進。

Roth’s theorem on arithmetic progressions is a result in additive combinatorics which states that if there is a set of positive integers that contains no non-trivial 3-term arithmetic progression, then we can give an estimate of the size of this set. There have been many refinements following Roth’s approach. In this paper, we will introduce the proofs of two refinements given by Bourgain and Sanders.