ON IMPROVING ROTH’S THEOREM IN THE PRIMES

Let $A\subset \{1,\dots ,N\}$ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density ${\it\alpha}=|A|/{\it\pi}(N)$, where ${\it\pi}(N)$ denotes the number of primes in the set $\{1,\dots ,N\}$. By modifying Helfgott and De Roton’s work [Imp...

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Veröffentlicht in:	Mathematika 2015-01, Vol.61 (1), p.49-62
1. Verfasser:	Naslund, Eric
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container_title	Mathematika
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description	Let $A\subset \{1,\dots ,N\}$ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density ${\it\alpha}=\|A\|/{\it\pi}(N)$, where ${\it\pi}(N)$ denotes the number of primes in the set $\{1,\dots ,N\}$. By modifying Helfgott and De Roton’s work [Improving Roth’s theorem in the primes. Int. Math. Res. Not. IMRN2011 (4) (2011), 767–783], we improve their bound and show that $$\begin{eqnarray}{\it\alpha}\ll \frac{(\log \log \log N)^{6}}{\log \log N}.\end{eqnarray}$$
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title	ON IMPROVING ROTH’S THEOREM IN THE PRIMES
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