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Let [0;a₁(x),a₂(x),…] be the regular continued fraction expansion of an irrational x ∈ [0,1]. We prove mainly that, for α > 0, β ≥ 0 and for almost all x ∈ [0,1], $li{m}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=\alpha /log2$ if α < 1 and β ≥ 0, $li{m}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=1/log2$ if α = 1 and β < 1, and, if α > 1 or α = 1 and β >1, $lim in{f}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=1/log2$, $lim su{p}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=\infty$, where $a{ⁿ}_i\left(x\right)=a_i\left(x\right)$ if $a_i\left(x\right)\le n^{\alpha }lo{g}^{\beta }n$ and $a{ⁿ}_i\left(x\right)=0$ otherwise, for all i ∈ 1,…,n.

Treating cancer patients with metastatic disease remains an ultimate challenge in clinical oncology. Because invasive cancer precludes or limits the use of surgery, metastatic setting is often associated with (poor) survival, rather than sustained remission, in patients with common cancers like lung, digestive or breast carcinomas. Mathematical modeling may help us better identify non detectable metastatic status to in turn optimize treatment for...