Roth's Theorem

There are various ways to state Roth's Theorem.

Roth's Statement

In Roth's seminal paper, On Certain Sets of Integers, Roth stated his theorem as follows:

Roth's Theorem

Let $A(x)$ denote the size of the largest subset of $\{1, 2, \ldots, x\}$ which avoids three-term arithmetic progressions. Then $\frac{A(x)}{x} = O \left( \frac{1}{\log \log x} \right).$

Alternative Statement: Density

We can also restate this theorem in a density form:

Roth's Theorem (Density)

Let $\delta \in (0,1]$. Then there exists a positive integer $N = N(\delta)$ such that for every subset of $\{1, 2, \ldots, N\}$ of size at least $\delta N$ contains an arithmetic progression of length three.

Alternative Statement: