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NEW BOUNDS FOR SZEMERÉDI’S THEOREM, III: A POLYLOGARITHMIC BOUND FOR $r_{4}(N)$

  • Ben Green (a1) and Terence Tao (a2)
Abstract

Define $r_{4}(N)$ to be the largest cardinality of a set $A\subset \{1,\ldots ,N\}$ that does not contain four elements in arithmetic progression. In 1998, Gowers proved that

$$\begin{eqnarray}r_{4}(N)\ll N(\log \log N)^{-c}\end{eqnarray}$$
for some absolute constant $c>0$ . In 2005, the authors improved this to
$$\begin{eqnarray}r_{4}(N)\ll N\text{e}^{-c\sqrt{\log \log N}}.\end{eqnarray}$$
In this paper we further improve this to
$$\begin{eqnarray}r_{4}(N)\ll N(\log N)^{-c},\end{eqnarray}$$
which appears to be the limit of our methods.

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References
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Mathematika
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