On a theorem of Erdos and Szemeredi
Résumé
K. F. Roth proved in 1957 that if $1 = q_1 < q_2 \!\!> h,$ where $h \geq x^{\theta}.$
Refining some of Szemeredi's ideas, it is proved in this paper that
%if 0 < < 1, and $\sum\frac{1}{b_i}<\infty$, then
$$Q(x+h) - Q(x) >\!\!> h,$$
where $x\geq h \geq x^{\theta}$ and $\theta >\frac{1}{2}$ is any constant.
%In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.
Domaines
Mathématiques [math]| Origine | Accord explicite pour ce dépôt |
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