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Dmitriy Frolenkov

Publications and source records attributed to Dmitriy Frolenkov.

6 recordsLinked to original sources
arXiv2013

A reinforcement of the Bourgain-Kontorovich's theorem by elementary methods II

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$ being bounded by an absolute constant $A.$ Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 has positive proportion in $\N.$ In this paper,using only elementary methods, the same theorem is proved with A=5.

math.NT↗
arXiv2012

A note on the reinforcement of the Bourgain-Kontorovich's theorem

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ whose partial quotients $d_1,d_2,...,d_{k}$ belong to a finite alphabet $\A\subseteq\N.$ In this paper it is proved for an alphabet $\A,$ such that the Hausdorff dimension $δ_{\A}$ of the set of infinite continued fractions whose partial quotients belong to $\A,$ that the set of numbers $d,$ satisfying Zaremba's conjecture with the alphabet $\A,$ has positive proportion in $\N.$ The result improves our previous reinforcement of the corresponding Bourgain-Kontorovich's theorem.

math.NT↗
arXiv2012

A reinforcement of the Bourgain-Kontorovich's theorem

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$ being bounded by an absolute constant $A.$ Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 has positive proportion in $\N.$ In this paper the same theorem is proved with A=7.

math.NT↗
arXiv2012

A reinforcement of the Bourgain-Kontorovich's theorem by elementary methods

Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 has positive proportion in $\N.$ The proof of this theorem is rather complicated and refers to the spectral theory. In this paper,using only elementary methods, the same theorem is proved with A=13 .

math.NT↗
arXiv2011

On a problem of Dobrowolski--Williams

In this paper we prove new upper bounds for the sum $\sum_{n=a+1}^{a+N}f(n)$, for a certain class of arithmetic functions $f$. Our results improve the previous results of G. Bachman and L. Rachakonda.

math.NT↗