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Integer sets containing no arithmetic progressions. (English) Zbl 0589.10062
Let \({\mathcal A}\subseteq [1,N]\cap {\mathbb{N}}\) contain no non-trivial 3-term arithmetic progression. It was shown by K. F. Roth [J. Lond. Math. Soc. 28, 104-109 (1953; Zbl 0050.04002)] that \(\#{\mathcal A}=O(N/\log \log N).\) The present paper proves that there exists a constant \(c>0\) such that \(\#{\mathcal A}=O(N/(\log N)^ c).\) A similar result has been obtained independently by Szemerédi. The proof follows Roth, and uses the Hardy- Littlewood circle method. The crucial new ingredient is the following variant of the large sieve. One takes \(a_ n\in {\mathbb{C}}\) (1\(\leq n\leq N)\) and \(S(x)=\sum a_ n e(nx).\) Define \(\xi (M)=(1/M) \max (\sum_{n\in {\mathcal P}}| a_ n|)\) where \({\mathcal P}\) runs over all arithmetic progressions of length M. Let \(x_ 1,...,x_ k\in {\mathbb{R}}\) satisfy the spacing condition \(\| x_ i-x_ j\| \geq \delta\) for \(i\neq j\). Then \[ \sum^{k}_{1}| S(x_ j)|^ 2\quad \ll \quad (N+\delta^{-1})\quad \xi (N^{1/k})\quad \sum | a_ n|. \] In certain situations this gives a saving relative to the usual form of the large sieve. The proof incorporates an idea of Szemerédi into Gallagher’s derivation of the large sieve.

11B25 Arithmetic progressions
11N35 Sieves
11P55 Applications of the Hardy-Littlewood method
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