Roth’s Theorem in the Primes
نویسنده
چکیده
We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled Hardy-Littlewood majorant property. We derive this from a rather more general result which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes.
منابع مشابه
Improving Roth’s theorem in the primes
Let A be a subset of the primes. Let δP (N) = |{n ∈ A : n ≤ N}| |{n prime : n ≤ N}| . We prove that, if δP (N) ≥ C log log logN (log logN)1/3 for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N ] contains a non-trivial three-term arithmetic progression. This improves on Green’s result [Gr], which needs δP (N) ≥ C s log log log log logN log log log logN .
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