Geodesic
A strengthening the one of a theorem of Bourgain -- Kontorovich
Dalʹnevostočnyj matematičeskij žurnal, Tome 20 (2020) no. 2, pp. 164-190

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The following result is proved in this work. Consider a set of $\mathfrak D_N $ not surpassing the $N$ of the denominators of those ultimate chain fractions, all incomplete private which belong to the alphabet $1,2,3,5$. Then inequality is fulfilled $|\mathfrak{D}_N|\gg N^{0.99}$. The calculation, made on a similar Burgeyin theorem – Of Kontorovich 2011, gives the answer $\mathfrak D_N \gg N^{0.80}$. 
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     author = {I. D. Kan},
     title = {A strengthening the one of a theorem of {Bourgain} -- {Kontorovich}},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {164--190},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2020_20_2_a5/}
}
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I. D. Kan. A strengthening the one of a theorem of Bourgain -- Kontorovich. Dalʹnevostočnyj matematičeskij žurnal, Tome 20 (2020) no. 2, pp. 164-190. http://geodesic.mathdoc.fr/item/DVMG_2020_20_2_a5/