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 |
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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) |
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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:
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