×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI