Geodesic
On Analogues of Heilbronn's Theorem
Matematičeskie zametki, Tome 111 (2022) no. 6, pp. 819-834.

Voir la notice de l'article provenant de la source Math-Net.Ru

Continued fractions with rational partial quotients arise in a natural way in the course of applying any $k$-ary gcd algorithm to the ratio of natural numbers $a$, $b$. The paper deals with the problem of estimating the average length of continued fractions of four types with rational partial quotients obtained by using Sorenson's right and left-shift $k$-ary gcd algorithms. This problem is reduced to the problem of estimating the number of solutions of an equation of special form with constraints on the variables and, in two cases, the number of solutions of a system of equations with constrained variables must be estimated.
Keywords: $k$-ary gcd algorithm, continued fractions with rational partial quotients
Mots-clés : continuants. 
@article{MZM_2022_111_6_a1,
     author = {D. A. Dolgov},
     title = {On {Analogues} of {Heilbronn's} {Theorem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {819--834},
     publisher = {mathdoc},
     volume = {111},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a1/}
}
TY  - JOUR
AU  - D. A. Dolgov
TI  - On Analogues of Heilbronn's Theorem
JO  - Matematičeskie zametki
PY  - 2022
SP  - 819
EP  - 834
VL  - 111
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a1/
LA  - ru
ID  - MZM_2022_111_6_a1
ER  - 
%0 Journal Article
%A D. A. Dolgov
%T On Analogues of Heilbronn's Theorem
%J Matematičeskie zametki
%D 2022
%P 819-834
%V 111
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a1/
%G ru
%F MZM_2022_111_6_a1
D. A. Dolgov. On Analogues of Heilbronn's Theorem. Matematičeskie zametki, Tome 111 (2022) no. 6, pp. 819-834. http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a1/

[1] H. Heilbronn, “On the average length of a class of finite continued fractions”, Number Theory and Analysis, Plenum, New York, 1969, 86–97 | MR

[2] G. Lochs, “Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmäigen Kettenbrüche”, Monatsh. Math., 65:1 (1961), 27–52 | DOI | MR | Zbl

[3] J. Sorenson, The k-Ary GCD Algorithm, Technical Report CS-TR-90-979, University of Wisconsin, Madison, WI, 1990

[4] A. Thue, “Et par antydninger til en taltheoretisk metode”, Kra. Vidensk. Selsk. Forh, 7 (1902), 57–75

[5] J. Solymosi, On the Thue–Vinogradov Lemma, 2021, arXiv: 2006.12319

[6] L. Aubry, “Un théoréme d'arithmétique”, Mathesis (4), 3 (1913)

[7] I. M. Vinogradov, “On a general theorem concerning the distribution of the residues and non-residues of powers”, Trans. Amer. Math. Soc., 29 (1927), 209–217 | DOI | MR | Zbl

[8] B. Straustrup, Programmirovanie: printsipy i praktika ispolzovaniya C++, Vilyams, M., 2011

[9] D. A. Dolgov, “O kontinuantakh tsepnykh drobei s ratsionalnymi nepolnymi chastnymi”, Diskret. matem., 2022 (to appear)

[10] J. Sorenson, “Two fast GCD Algorithms”, J. Algorithms, 16 (1994), 110–144 | DOI | MR | Zbl

[11] D. A. Dolgov, “O tsepnykh drobyakh s ratsionalnymi nepolnymi chastnymi”, Chebyshevskii sb., 2022 (to appear)

[12] A. A. Karatsuba, Osnovy analiticheskoi teorii chisel, Nauka, M., 1983 | MR