Far Eastern Mathematical Journal

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A strengthening the one of a theorem of Bourgain-Kontorovich


Kan I.D.

2020, issue 2, Ñ. 164–190
DOI: https://doi.org/10.47910/FEMJ202018


Abstract
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 $\mathfrak D_N \gg N^{0.80}$.

Keywords:
continued fraction, exponensional sum, Zaremba conjecture

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