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ON IMPROVING ROTH’S THEOREM IN THE PRIMES

Part of: Sequences and sets Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  09 October 2014

Eric Naslund
Eric Naslund*
Affiliation:
Princeton University Mathematics Department, Fine Hall Room 304, Princeton, NJ 08544-1044, USA email naslund@math.princeton.edu
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Abstract

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

$$\begin{eqnarray}{\it\alpha}\ll \frac{(\log \log \log N)^{6}}{\log \log N}.\end{eqnarray}$$

MSC classification

Primary: 11B25: Arithmetic progressions 11N13: Primes in progressions 11N36: Applications of sieve methods 11P32: Goldbach-type theorems; other additive questions involving primes 11B30: Arithmetic combinatorics; higher degree uniformity
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 49 - 62
Copyright
Copyright © University College London 2014 

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