Erdős Semi-Groups, Arithmetic Progressions, and Szemerédi's Theorem

In this paper, we introduce and study a certain type of sub semigroup of R/Z which turns out to be closely related to Szemerédi's theorem on arithmetic progressions. Two motivating problemsSzemerédi's theorem on arithmetic progressions is perhaps one of the most interesting topics in mathematics.There are a lot of materials on this topic, for more details see [14], [8] and [9].One of the reasons for Szemerédi's theorem being popular is that it has several proofs with very different backgrounds.The aim of this paper is to introduce another point of view for Szemerédi's theorem.For fractal dimensions and arithmetic structures, there are some recent results, see for example [5], [7].Here we adopt a different but related approach.We will study the set of numbers in [0, 1) whose binary digit expansion does not have arbitrarily long arithmetic progressions of positions of digit 1.Definition 1.1.We say x ∈ [0, 1) is Erdősian, if the binary expansion of x does not contain arbitrarily long arithmetic progressions of positions of digit 1.The collection of all Erdősian numbers is a subset of [0, 1) and we call it the Erdős set E.