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# On a theorem of Erdos and Szemeredi

Abstract : 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.
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Cited literature [8 references]

https://hal.archives-ouvertes.fr/hal-01103868
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Mangala J Narlikar. On a theorem of Erdos and Szemeredi. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1980, Volume 3 - 1980, pp.41 - 47. ⟨10.46298/hrj.1980.90⟩. ⟨hal-01103868⟩

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