On triples in arithmetic progression (Q1964140)

It was shown by \textit{K. F. Roth} [J. Lond. Math. Soc. 28, 104-109 (1953; Zbl 0050.04002)] that a set \(A\) of positive integers \(n\leq x\) containing \(\gg x(\log\log x)^{-1}\) elements, must contain three distinct terms in arithmetic progression. The result was sharpened by the reviewer [J. Lond. Math. Soc. (2) 35, 385-394 (1987; Zbl 0589.10062)] and \textit{E. Szemerédi} [Acta Math. Hung. 56, 155-158 (1990; Zbl 0721.11007)], to show that \(\gg x(\log x)^{-c}\) terms suffice, for a certain positive constant \(c\). The present paper shows that in fact it is enough for \(A\) to contain \(\gg x(\log\log x/\log x)^{1/2}\) terms. The author uses a form of the circle method, but instead of examining the behaviour of \(A\) in arithmetic progressions, one uses `Bohr sets' defined by conditions of the form \(\|n\theta_j\|< \delta\) for \(j\leq J\).