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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.

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