$A_2$-continued fraction representation of real numbers and its geometry
We study the geometry of representations of numbers by continued fractions whose elements belong to the set $A_2 = {α_1, α_2}$ ($A_2$-continued fraction representation). It is shown that, for $α_1 α_2 ≤ 1/2$, every point of a certain segment admits an $A_2$-continued fraction representation. Moreove...
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| Date: | 2009 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2009
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3033 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509051788460032 |
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| author | Dmytrenko, S. O. Kyurchev, D. V. Pratsiovytyi, M. V. Дмитренко, С. O. Кюрчев, Д. В. Працьовитий, М. В. |
| author_facet | Dmytrenko, S. O. Kyurchev, D. V. Pratsiovytyi, M. V. Дмитренко, С. O. Кюрчев, Д. В. Працьовитий, М. В. |
| author_sort | Dmytrenko, S. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:50Z |
| description | We study the geometry of representations of numbers by continued fractions whose elements belong to the set $A_2 = {α_1, α_2}$ ($A_2$-continued fraction representation). It is shown that, for $α_1 α_2 ≤ 1/2$, every point of a certain segment admits an $A_2$-continued fraction representation. Moreover, for $α_1 α_2 = 1/2$, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose $A_2$-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its $A_2$-continued fraction representation form a homogeneous Markov chain are also investigated. |
| first_indexed | 2026-03-24T02:34:58Z |
| format | Article |
| fulltext |
UDK 511.72
S. O. Dmytrenko, D. V. Kgrçev (Nac. ped. un-t, Ky]v),
M. V. Prac\ovytyj (Nac. ped. un-t, In-t matematyky NAN Ukra]ny, Ky]v)
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL
TA JOHO HEOMETRIQ
We study the geometry of the representation of real numbers in terms of continued fractions whose
elements take values from the set A2 1 2= { , }α α ( A2 -continued fraction representation). For the case
α α1 2 1 2≤ / , we prove that any point of some closed interval has a A2 -continued fraction representati-
on, and that, in the case α α1 2 1 2= / , the representation is unique, except for a countable set of points.
In the latter case, we establish the basic metric relation, describe metric properties of sets of numbers
whose A2 -continued fraction representation does not contain a given combination of two elements, and
investigate properties of the random variable whose A2 -continued fraction representation elements form
a homogeneous Markov chain.
Yzuçaetsq heometryq predstavlenyq çysel cepn¥my drobqmy, πlement¥ kotor¥x prynadleΩat
mnoΩestvu A2 1 2= { , }α α (cepnoe A2 -predstavlenye). Dokazano, çto pry α α1 2 1 2≤ / kaΩ-
daq toçka opredelennoho otrezka ymeet cepnoe A2 -predstavlenye, pryçem pry α α1 2 1 2= /
predstavlenye edynstvennoe, za ysklgçenyem sçetnoho mnoΩestva toçek. Dlq posledneho slu-
çaq najdeno osnovnoe metryçeskoe sootnoßenye, opysan¥ metryçeskye svojstva mnoΩestva çy-
sel, cepnoe A2 -predstavlenye kotor¥x ne soderΩyt zadannoj kombynacyy dvux πlementov, a
takΩe yzuçen¥ svojstva sluçajnoj velyçyn¥, πlement¥ cepnoho A2 -predstavlenyq kotoroj
obrazugt odnorodnug cep\ Markova.
1. Vstup. Isnu[ bahato riznyx sposobiv zobraΩennq dijsnyx çysel i vidpovidnyx
]m metryçnyx ta jmovirnisnyx teorij. Iz koΩnym iz nyx pov�qzana svoq systema
cylindryçnyx mnoΩyn (cylindriv), wo utvorggt\ systemu podribnggçyx roz-
byttiv vidrizka i porodΩugt\ svog vlasnu heometrig, na qkij ©runtu[t\sq met-
ryçna teoriq. Odni zobraΩennq vykorystovugt\ skinçennyj alfavit (nabir
cyfr), inßi � neskinçennyj. Do perßyx vidnosqt\sq zobraΩennq çysel s-
adyçnymy drobamy, Q-zobraΩennq [1] ta in. Sered zobraΩen\ iz neskinçennym
alfavitom najbil\ß vidomymy [ zobraΩennq dijsnyx çysel rqdamy Ostrohrad-
s\koho (1- ta 2-ho vydiv) [2], rqdamy Lgrota [3] towo. Do ostannix naleΩyt\ i
sposib zobraΩennq dijsnyx çysel elementarnymy lancghovymy drobamy, ele-
mentamy qkyx [ natural\ni çysla. Osnovy metryçno] teori] c\oho zobraΩennq
bulo zakladeno we na poçatku 20-ho stolittq v robotax O. Xinçyna [4], P. Levi
[5] ta in. U naukovyx doslidΩennqx s\ohodni fihurugt\ takoΩ lancghovi roz-
klady, vidminni vid elementarnyx, sered qkyx taki, wo vykorystovugt\ skin-
çennyj alfavit. Ce lancghovi rozklady DanΩua [6], Lexnera [7], Fareq [8]
ta in.
Danu robotu prysvqçeno novomu sposobu zobraΩennq dijsnyx çysel lancg-
hovym drobom iz dvoelementnym alfavitom A2 = { , }α α1 2 , 0 < α1 < α2. Roz-
hlqda[t\sq mnoΩyna LA2
vsix neskinçennyx lancghovyx drobiv vyhlqdu
1
1
11
2
a
a
an
+
+ …+ + …
,
elementy an qkyx naleΩat\ A2 (taki lancghovi droby nazyvatymemo lancg-
hovymy A2 -drobamy). Qkwo α1 i α2 [ natural\nymy, to vidomo [1, c. 248], wo
LA2
[ nide ne wil\nog mnoΩynog nul\ovo] miry Lebeha. Dewo nespodivanym
vyqvyvsq fakt, wo dostatn\o vykonannq umovy α α1 2 = 1 2/ dlq toho, wob mno-
Ωyna LA2
bula vidrizkom, bud\-qku toçku qkoho (okrim zçyslenno] mnoΩyny)
© S. O. DMYTRENKO, D. V. KGRÇEV, M. V. PRAC|OVYTYJ, 2009
452 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL … 453
moΩna zobrazyty lancghovym A2 -drobom [dynym sposobom.
Take zobraΩennq çysel za suttg [ lancghovym, za formog ma[ vlastyvosti
dvijkovoho koduvannq dijsnyx çysel, oskil\ky alfavit mistyt\ dva symvoly.
Ale heometriq lancghovoho A2 -zobraΩennq ma[ pryncypovi vidminnosti qk vid
dvijkovoho zobraΩennq, tak i vid zobraΩennq çysel elementarnymy lancghovy-
my drobamy, wo vidobraΩeno u vlastyvostqx cylindryçnyx mnoΩyn. U cij ro-
boti my budu[mo osnovy metryçno] teori] c\oho zobraΩennq i doslidΩu[mo vlas-
tyvosti vypadkovoho lancghovoho A2 -drobu, elementy qkoho utvorggt\ odno-
ridnyj lancgh Markova.
2. Lancghovi A2-droby. Neskinçennyj lancghovyj drib
a
a
a
0
1
2
1
1+
+
+ …
(1)
budemo zobraΩaty (formal\no zapysuvaty) u vyhlqdi [ ]; , , , ,a a a an0 1 2 … … , a
skinçennyj
a
a
a
an
0
1
2
1
1
1
+
+
+ …+
(2)
� u vyhlqdi [ ]; , , ,a a a an0 1 2 … . Qkwo a0 = 0, to lancghovyj drib (1) zobraΩa-
tymemo u vyhlqdi [ ], , , ,a a an1 2 … … , a lancghovyj drib (2) � u vyhlqdi [ ,a1
a an2, , ]… . My takoΩ vykorystovuvatymemo zapys lancghovoho drobu [ ; ,a a0 1
a an2, , , ]… … u vyhlqdi [ ]; , , , ,a a a a rn n0 1 2 1… − , de rn = [ ]; , , ,a a an n n k+ +… …1 �
n -j zalyßok poçatkovoho lancghovoho drobu. Periodyçnyj lancghovyj drib
[ ], , , , , , , , , , , ,a a a c c c c c cn k k1 2 1 2 1 2… … … …
budemo zapysuvaty qk [ ], , , , ( , , , )a a a c c cn k1 2 1 2… … , de ( , , , )c c ck1 2 … � period da-
noho lancghovoho drobu.
Nexaj A2 1 2= { , }α α , 0 < α1 < α2, � zadana mnoΩyna dijsnyx çysel.
Neskinçennyj lancghovyj drib vyhlqdu [ ]; , , , ,a a a an0 1 2 … … , de a0 = 0,
a An ∈ 2 , n = 1, 2, … , nazyvatymemo lancghovym A2 -drobom. KoΩen lancg-
hovyj A2 -drib [ zbiΩnym, oskil\ky vykonu[t\sq kryterij zbiΩnosti [4, c. 17]:
rqd ann =
∞∑ 1
rozbiha[t\sq.
Poznaçymo
β1 = [ ]( , )α α2 1 =
α α α α α α
α
1
2
2
2
1 2 1 2
2
4
2
+ −
, (3)
β2 = [ ]( , )α α1 2 =
α α α α α α
α
1
2
2
2
1 2 1 2
1
4
2
+ −
. (4)
Z oznaçen\ β1 i β2 ma[mo
β2 =
1
1 1α β+
i β2 =
α
α
β2
1
1. (5)
Z (5) vyplyva[
β
α
2
2
=
β
α
1
1
= λ > 0, β β2 1− = λ α α( )2 1− .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
454 S. O. DMYTRENKO, D. V. KGRÇEV, M. V. PRAC|OVYTYJ
Poznaçymo çerez LA2
mnoΩynu vsix lancghovyx A2 -drobiv, tobto
LA2
= x x a a a a A nn n: , , , , ; , , ,[ ]= … … ∈ = …{ }1 2 2 1 2 .
Todi oçevydno, wo
min LA2
= inf LA2
= β1, max LA2
= sup LA2
= β2 ,
LA2
⊆ [ ],β β1 2 = [ ],λα λα1 2 .
Nas cikavqt\ topoloho-metryçni vlastyvosti mnoΩyny LA2
.
MoΩlyvi try vypadky:
1) α α2 1− = β β2 1− ⇔ α β2 1+ = α β1 2+ ;
2) α α2 1− > β β2 1− ⇔ α β2 1+ > α β1 2+ ;
3) α α2 1− < β β2 1− ⇔ α β2 1+ < α β1 2+ .
Z uraxuvannqm vyraziv (3) i (4) rivnist\ α α2 1− = β β2 1− rivnosyl\na rivnosti
α α2 1− =
α α α α α α
α
α α α α α α
α
1
2
2
2
1 2 1 2
1
1
2
2
2
1 2 1 2
2
4
2
4
2
+ − − + −
,
α α α α1
2
2
2
1 24+ = 3 1 2α α ,
α α1 2 =
1
2
.
U c\omu vypadku β1 =
1
2 2α
= α1, β2 =
1
2 1α
= α2 .
ZauvaΩymo, wo okremo] uvahy zasluhovu[ pidvypadok α1 = β1 = 1 2/ , α2 =
= β2 = 1. Umova α α2 1− < β β2 1− rivnosyl\na umovi 0 < α α1 2 < 1 2/ , a
umova α α2 1− > β β2 1− � nerivnosti α α1 2 > 1 2/ .
3. Pidxidni droby. Nahada[mo [4], wo pidxidnym drobom porqdku n danoho
lancghovoho drobu
[ ]; , , , ,a a a an0 1 2 … …
nazyva[t\sq çyslo
p
q
n
n
, wo [ znaçennqm skinçennoho lancghovoho drobu
[ ]; , , ,a a a an0 1 2 … , tobto n -ho vidrizka danoho lancghovoho drobu. Pry c\omu
çyslo pn nazyva[t\sq çysel\nykom pidxidnoho drobu, a qn � znamennykom.
Vidomyj [4] zakon utvorennq pidxidnyx drobiv formulg[t\sq tak: dlq dovil\-
noho natural\noho n ≥ 2
pn = a p pn n n− −+1 2 ,
qn = a q qn n n− −+1 2,
pry c\omu vvaΩa[t\sq, wo p0 = a0, q0 = 1, p1 = a a1 0 1+ , q1 = a1.
Oskil\ky pn i qn zaleΩat\ vid perßyx n + 1 elementiv lancghovoho
drobu, to budemo poznaçaty ]x takoΩ qk p a a an( , , , )0 1 … i q a a an( , , , )0 1 … vidpo-
vidno ( abo p a an( , , )1 … i q a an( , , )1 … pry a0 = 0 ).
Dlq pidxidnyx drobiv spravedlyvymy [ nastupni tverdΩennq [4].
Teorema 1. 1. Dlq bud\-qkoho k ∈ N q p p qk k k k− −−1 1 = ( )−1 k .
2. Dlq bud\-qkoho k ∈ N
p
q
p
q
k
k
k
k
−
−
−1
1
=
( )−
−
1
1
k
k kq q
.
3. Dlq bud\-qkoho k ∈ N q p p qk k k k− −−2 2 = ( )− −1 1k
ka .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL … 455
4. Dlq bud\-qkoho k ≥ 2
p
q
p
q
k
k
k
k
−
−
−2
2
=
( )− −
−
1 1
2
k
k
k k
a
q q
.
5. Dlq bud\-qkoho k ∈ N
q
q
k
k−1
= [ ]; , ,a a ak k− …1 1 .
6. Dlq bud\-qkoho k ∈ { 1, 2, … , n } [ ]; , , ,a a a an0 1 2 … =
p r p
q r q
k k k
k k k
− −
− −
+
+
1 2
1 2
.
Lema 1 [9]. Znamennyk qn pidxidnoho drobu porqdku n periodyçnoho lan-
cghovoho drobu [( )]c obçyslg[t\sq za formulog
qn =
1
4
4
2
4
22
2 1 2 1
c
c c c c
n n
+
+ +
− − +
+ +
.
Naslidok 1. Dlq znamennykiv qn pidxidnyx drobiv lancghovyx A2 -drobiv
pry koΩnomu natural\nomu n magt\ misce toçni ocinky
qn ≥
1
4
4
2
4
21
2
1 1
2 1
1 1
2 1
α
α α α α
+
+ +
− − +
+ +n n
,
(6)
qn ≤
1
4
4
2
4
22
2
2 2
2 1
2 2
2 1
α
α α α α
+
+ +
− − +
+ +n n
.
Dovedennq. Oçevydno, wo pry koΩnomu n = 1, 2, … vykonu[t\sq neriv-
nist\
q
n
( , , , )α α α1 1 1…
1 244 344
≤ q a a an( , , , )2 2 … ≤
q
n
( , , , )α α α2 2 2…
1 244 344
.
Skorystavßys\ lemog 1, otryma[mo (6).
Naslidok 2. Qkwo α1 = 1 2/ , α2 = 1, to dlq znamennykiv pidxidnyx dro-
biv magt\ misce nerivnosti
qn ≥
2 17
17
1 17
4
1 17
4
1 1+
− −
+ +n n
,
qn ≤
5
5
1 5
2
1 5
2
1 1+
− −
+ +n n
, n ∈ N .
4. Cylindryçni mnoΩyny i ]x vlastyvosti. Cylindrom ranhu m z osno-
vog c c cm1 2 … nazyvatymemo mnoΩynu ′ …∆c c cm1 2
vsix x LA∈
2
, qki magt\ lan-
cghove A2 -zobraΩennq z perßymy m elementamy, wo dorivnggt\ vidpovidno
c c cm1 2, , ,… , tobto
′ …∆c cm1
= x x c c c a a a Am m m m i: , , , , , , ,= … …[ ] ∈{ }+ + +1 2 1 2 2 .
Oçevydno, wo
min ′∆c1
= c1
2
1
,
β
= c1 2+[ ]β , min ′∆c c1 2
= c c1 2
1
1
, ,
β
= c c1 2 1, +[ ]β ,
max ′∆c1
= c1
1
1
,
β
= c1 1+[ ]β , max ′∆c c1 2
= c c1 2
2
1
, ,
β
= c c1 2 2, +[ ]β ,
i v zahal\nomu vypadku
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
456 S. O. DMYTRENKO, D. V. KGRÇEV, M. V. PRAC|OVYTYJ
min ′ …∆c cm1
=
c c c c c m
c c c c c m
m m m
m m m
1
2
1 1 2
1
1
1 1 1
1
1
, , , , , , ,
, , , , , , ,
…
= … +[ ]
…
= … +[ ]
−
−
β
β
β
β
pry neparnomu
pry parnomu
max ′ …∆c cm1
=
c c c c c m
c c c c c m
m m m
m m m
1
1
1 1 1
1
2
1 1 2
1
1
, , , , , , ,
, , , , , , .
…
= … +[ ]
…
= … +[ ]
−
−
β
β
β
β
pry neparnomu
pry parnomu
Vidrizok z kincqmy min ′ …∆c cm1
i max ′ …∆c cm1
nazyvatymemo cylindryçnym
vidrizkom ranhu m z osnovog c cm1 … i poznaçatymemo ∆c cm1… . Interval, kinci
qkoho zbihagt\sq z kincqmy ∆c cm1… , poznaçymo ∇ …c cm1
i nazyvatymemo cylind-
ryçnym intervalom ranhu m z osnovog c c cm1 2 … .
Oçevydno, wo ′ …∆c cm1
⊂ ∆c cm1… , ale ne zavΩdy ′ …∆c cm1
= ∆c cm1… . Neob-
xidni i dostatni umovy dlq ostann\o] rivnosti budut\ navedeni dali.
Dlq dovil\noho natural\noho m magt\ misce nastupni vlastyvosti:
1. ′ …∆c c cm1
⊂ ′ …∆c cm1
. Bil\ß toho, ′ …∆c cm1
= ′ ′… …∆ ∆c c c cm m1 1 1 2α αU .
Cq vlastyvist\ vyplyva[ bezposeredn\o z oznaçennq cylindra.
2. ∆c c cm1… ⊂ ∆c cm1… , ale, vzahali kaΩuçy,
∆c cm1… ≠
∆ ∆c c c cm m1 1 1 2… …α αU .
Vklgçennq [ oçevydnym, a nerivnist\ ma[ misce, zokrema, koly A2 = { , }1 3 :
∆1 ≠ ∆ ∆11 13U .
3 . inf ∆c cm1 1… α < inf ∆c cm1 2… α pry neparnomu m i inf ∆c cm1 1… α >
> inf ∆c cm1 2… α pry parnomu m.
Spravdi, pry parnomu m
inf ∆c cm1 1… α = c cm1 1 2, , ,… +[ ]α β < c cm1 2 2, , ,… +[ ]α β = inf ∆c cm1 2… α ,
a pry neparnomu m
inf ∆c cm1 1… α = c cm1 1 1, , ,… +[ ]α β < c cm1 2 1, , ,… +[ ]α β = inf ∆c cm1 2… α .
4. Qkwo α α2 1− = β β2 1− , wo rivnosyl\no α β2 1+ = α β1 2+ , to
∆ ∆c c c cm m1 1 1 2… …α αI = c cm1 1 2, , ,… +[ ]α β = c cm1 2 1, , ,… +[ ]α β .
Dijsno, pry parnomu m
max ∆c cm1 2… α = c cm1 2 1, , ,… +[ ]α β = c cm1 1 2, , ,… +[ ]α β = min ∆c cm1 1… α ,
a pry neparnomu m
max ∆c cm1 1… α = c cm1 1 2, , ,… +[ ]α β = c cm1 2 1, , ,… +[ ]α β = min ∆c cm1 2… α .
5. Qkwo α α2 1− < β β2 1− ⇔ α β2 1+ < α β1 2+ , to
∆ ∆c c c cm m1 1 1 2… …α αI = a b,[ ],
de
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL … 457
a =
c c m
c c m
m
m
1 1 2
1 2 1
, , , ,
, , , ,
… +[ ]
… +[ ]
α β
α β
pry parnomu
pry neparnomu
b =
c c m
c c m
m
m
1 2 1
1 1 2
, , , ,
, , , .
… +[ ]
… +[ ]
α β
α β
pry parnomu
pry neparnomu
Dovedennq. Nexaj m [ parnym. Todi
max ∆c cm1 2… α = c cm1 2 1, , ,… +[ ]α β > c cm1 1 2, , ,… +[ ]α β = min ∆c cm1 1… α .
Na pidstavi vlastyvosti 3 ma[mo
∆ ∆c c c cm m1 2 1 1… …α αI = min ; max∆ ∆c c c cm m1 1 1 2… …[ ]α α = a b,[ ].
Qkwo m [ neparnym, to
max ∆c cm1 1… α = c cm1 1 2, , ,… +[ ]α β > c cm1 2 1, , ,… +[ ]α β = min ∆c cm1 2… α .
U c\omu vypadku na pidstavi vlastyvosti 3 otrymu[mo
∆ ∆c c c cm m1 1 1 2… …α αI = min ; max∆ ∆c c c cm m1 2 1 1… …[ ]α α = a b,[ ].
6. Qkwo α α2 1− ≤ β β2 1− ⇔ α β2 1+ ≤ α β1 2+ , to
∆c cm1… =
∆ ∆c c c cm m1 1 1 2… …α αU .
Dana vlastyvist\ [ naslidkom vlastyvostej 4 i 5.
7. Qkwo α α2 1− > β β2 1− , wo rivnosyl\no α β2 1+ > α β1 2+ , to
∆ ∆c c c cm m1 1 1 2… …α αI = ∅ . (7)
Dovedennq. Nexaj m [ parnym. Todi
max ∆c cm1 2… α = c cm1 2 1, , ,… +[ ]α β < c cm1 1 2, , ,… +[ ]α β = min ∆c cm1 1… α .
U vypadku neparnoho m
max ∆c cm1 1… α = c cm1 1 2, , ,… +[ ]α β < c cm1 2 1, , ,… +[ ]α β = min ∆c cm1 2… α .
Na pidstavi vlastyvosti 3 otrymu[mo (7).
8. DovΩyna cylindryçnoho vidrizka ∆c cn1… obçyslg[t\sq za formulog
∆c cn1… =
β β
β β
2 1
1 1 2 1
−
+ +− −( )( )q q q qn n n n
. (8)
Spravdi,
∆c cn1… = max min∆ ∆c c c cn n1 1… …− =
= c c c cn n1 2
1
1 1
1, , , , , ,…[ ] − …[ ]− −β β =
=
β
β
β
β
2
1
1
2
1
1
1
1
1
1
1
1
−
−
−
−
−
−
−
−
+
+
− +
+
p p
q q
p p
q q
n n
n n
n n
n n
=
p p
q q
p p
q q
n n
n n
n n
n n
+
+
− +
+
−
−
−
−
β
β
β
β
2 1
2 1
1 1
1 1
=
=
β β
β β
2 1 1 1 1 1
1 1 2 1
( ) ( )
( )( )
p q q p p q q p
q q q q
n n n n n n n n
n n n n
− − − −
− −
− − −
+ +
=
=
( )
( )( )
β β
β β
2 1 1 1
1 1 2 1
− −
+ +
− −
− −
p q q p
q q q q
n n n n
n n n n
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
458 S. O. DMYTRENKO, D. V. KGRÇEV, M. V. PRAC|OVYTYJ
Vraxovugçy, wo p q q pn n n n− −−1 1 = ( )−1 n , otrymu[mo formulu (8).
Naslidok 3. ∆c cn1… → 0, n → ∞ .
Naslidok 4. Diametr cylindra ′ …∆c cm1
, zbihagçys\ iz dovΩynog ∆c cm1… ,
dorivng[ vyrazu (8).
Naslidok 5. Qkwo α1 = 1 2/ , a α2 = 1, to β1 = 1 2/ , β2 = 1 i
∆c cn1… =
1
21 1( )( )q q q qn n n n− −+ +
.
9. Osnovne metryçne vidnoßennq ma[ vyhlqd
∆
∆
c c c
c c
n
n
1
1
…
…
=
1 11
1
2
1
1
1
2
1
+
+
+ +
+ +
− −
− −
β β
β β
q
q
q
q
c
q
q
c
q
q
n
n
n
n
n
n
n
n
.
Dovedennq. Vykorystovugçy formulu (8) dlq vyraziv ∆c cn1… i ∆c c cn1…
i zakon utvorennq znamennykiv pidxidnyx drobiv, otrymu[mo
∆
∆
c c c
c c
n
n
1
1
…
…
=
( )( )
( )( )
q q q q
cq q q cq q q
n n n n
n n n n n n
+ +
+ + + +
− −
− −
β β
β β
1 1 2 1
1 1 1 2
=
=
1 11
1
2
1
1
1
2
1
+
+
+ +
+ +
− −
− −
β β
β β
q
q
q
q
c
q
q
c
q
q
n
n
n
n
n
n
n
n
.
10. Qkwo α α2 1− = β β2 1− (tobto α α1 2 = 1 2/ , α1 = β1, α2 = β2 ),
to
∆
∆
c c c
c c
n
n
1
1
…
…
=
1
2 1 2
1
2 1
+
+ +
−
−
c
q
q
c c
q
q
n
n
n
n
, (9)
∆
∆
c c
c c
n
n
1 1
1 2
…
…
α
α
=
1 2 1 2
1 2 1 2
1
1
2
2
2
1
2
1
1
2
1
1
+
+ +
+
+ +
− −
− −
α α α
α α α
q
q
q
q
q
q
q
q
n
n
n
n
n
n
n
n
. (10)
Dovedennq. Vykorystovugçy vlastyvist\ 9, otrymu[mo
λ1 =
∆
∆
c c
c c
n
n
1 1
1
…
…
α
=
1 11
1
2
1
1 1
1
1 2
1
+
+
+ +
+ +
− −
− −
β β
α β α β
q
q
q
q
q
q
q
q
n
n
n
n
n
n
n
n
.
Ale β1 = α1, β2 = α2, α1 = 1 2 2/( )α . Tomu pislq sprowennq ma[mo
λ1 =
1
2 1 2
1
1
1
2
1
1
+
+ +
−
−
α
α α
q
q
q
q
n
n
n
n
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL … 459
Rivnist\ (10) [ naslidkom rivnosti (9).
11. Qkwo α1 = 1 2/ , α2 = 1, to β1 = 1 2/ , β2 = 1 i
∆
∆
c c
c c
n
n
1
1
1
2…
…
−
=
2
3 2
1
1
+
+
−
−
q
q
q
q
n
n
n
n
,
∆
∆
c c
c c
n
n
1
1
1…
…
=
1
3 2
1
1
+
+
−
−
q
q
q
q
n
n
n
n
, (11)
∆
∆
c c
c c
n
n
1
1
1
2
1
…
…
−
=
2
1
1
1
+
+
−
−
q
q
q
q
n
n
n
n
= 1
1
1 1
+
+ −q
q
n
n
.
Dovedennq. Pidstavlqgçy znaçennq α1 i α2 v rivnosti (9), (10), pislq
sprowennq otrymu[mo rivnosti (11).
5. Topoloho-metryçni vlastyvosti mnoΩyny LA2
.
Teorema 2. Qkwo α α1 2 ≤ 1 2/ , to LA2
= β β1 2,[ ].
Dovedennq. Oskil\ky oçevydno, wo LA2 1 2⊂ [ ]β β, , to zalyßa[t\sq doves-
ty, wo β β1 2 2
,[ ] ⊂ LA .
Nexaj x � dovil\na toçka vidrizka β β1 2,[ ]. Zhidno z vlastyvistg 6 β β1 2,[ ] =
=
∆ ∆α α1 2
U . Tomu isnu[ c A1 2 1 2∈ = { }α α, take, wo x c∈∆
1
(zvyçajno, qkwo
x ∈∆ ∆α α1 2
I , to take c1 vyznaça[t\sq neodnoznaçno). Za ti[g Ω vlastyvistg
∆c1
=
∆ ∆c c1 1 1 2α αU . Tomu isnu[ c A2 2∈ take, wo x c c∈∆
1 2
i t. d. Qkwo
x c ck
∈ …∆
1
(a ce oznaça[, wo x = c c a ak k k1 1 2, , , , ,… …[ ]+ + , de ak j+ � deqki ele-
menty mnoΩyny A2 ), to z toho, wo
∆c ck1… = ∆ ∆c c c ck k1 1 1 2… …α αU ,
vyplyva[ isnuvannq c Ak+ ∈1 2 takoho, wo x c c ck k
∈ … +
∆
1 1
i t. d.
OtΩe, isnu[ neskinçenna poslidovnist\ ck{ }, c Ak ∈ 2 taka, wo x naleΩyt\
vsim cylindryçnym vidrizkam
∆c1
, ∆c c1 2
, … , ∆c c ck1 2… , … .
Oskil\ky, zhidno z vlastyvistg 2
∆c c ck k1 1… +
⊂ ∆c ck1…
i zhidno z naslidkom 3
∆c ck1… → 0, k → ∞ ,
to za aksiomog Kantora isnu[ [dyna toçka, qka naleΩyt\ vsim cym vidrizkam, a
takog [ toçka x . Tomu
x =
∆c c
k
k1
1
…
=
∞
I = c ck1, , ,… …[ ] ∈ LA2
,
wo j potribno bulo dovesty.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
460 S. O. DMYTRENKO, D. V. KGRÇEV, M. V. PRAC|OVYTYJ
Naslidok 6. Qkwo α α1 2 ≤ 1 2/ , to ′ …∆c cm1
= ∆c cm1… .
Teorema 3. Qkwo α α1 2 = 1 2/ , to zçyslenna mnoΩyna toçok x ∈[ ]β β1 2,
ma[ dva lancghovyx A2 -zobraΩennq, reßta Ω toçok magt\ [dyne zobraΩennq.
Dovedennq. Qkwo x =
∆ ∆c c c cm m1 1 1 2… …α αI , to x ma[ dva lancghovyx
A2 -zobraΩennq zhidno z vlastyvistg 6 cylindryçnyx mnoΩyn. I takyx toçok,
oçevydno, [ zçyslenna mnoΩyna (dlq koΩnoho natural\noho m ]x kil\kist\ [
skinçennog).
Qkwo x ne [ spil\nog toçkog cylindryçnyx vidrizkiv
∆a am1 1… α i ∆a am1 2… α
dlq Ωodnoho naboru a am1, ,… , to çysla ck , isnuvannq qkyx vstanovleno pry
dovedenni poperedn\o] teoremy, vyznaçagt\sq odnoznaçno, oskil\ky
∇ ∇… …− −c c c ck k1 1 1 1 1 2α αI = ∅ .
U c\omu vypadku dlq toçky x isnu[ [dyna poslidovnist\ ck{ } taka, wo x =
= c c ck1 2, , , ,… …[ ]. Spravdi, prypustymo, wo
x = c c ck1 2, , , ,… …[ ] = d d dk1 2, , , ,… …[ ].
Qkwo α1 = c1 ≠ d1 = α2 , to x ∈∆α1
i x ∈∆α2
, a ce supereçyt\ tomu, wo
∇ ∇α α1 2
I = ∅ i x ≠
∆ ∆α α1 2
I . OtΩe, c1 = d1 .
Nexaj teper ci = di , i = 1, m , i cm+1 ≠ dm+1. Todi x c cm
∈ …∆
1 1α i
x c cm
∈ …∆
1 2α , tobto x = ∆ ∆c c c cm m1 1 1 2… …α αI , wo supereçyt\ umovi
∇ ∇… …c c c cm m1 1 1 2α αI = ∅ ,
x ≠
∆ ∆c c c cm m1 1 1 2… …α αI .
OtΩe, dlq dovil\noho natural\noho m cm ≠ dm , wo j potribno bulo do-
vesty.
U vypadku α α1 2 = 1 2/ toçky x ∈[ ]β β1 2, , wo [ kincqmy deqkoho cylindra,
budemo nazyvaty A2 -racional\nymy. Do takyx toçok naleΩat\ ti, wo magt\
dva lancghovyx A2 -zobraΩennq, a takoΩ toçky β1 i β2 . Oçevydno, wo
koΩne A2 -racional\ne çyslo (okrim β1 i β2 ) moΩna podaty u vyhlqdi
x = a a an1 2 1 1 20
, , , , , ( , )…[ ]α α α
abo
x = a a an1 2 2 2 10
, , , , , ( , )…[ ]α α α ,
de n0 � najmenßyj nomer takyj, wo toçka x [ spil\nym kincem dvox cylindriv
ranhu n dlq vsix n > n0 .
Qkwo toçka x ne [ kincem Ωodnoho cylindriçnoho vidrizka ∆c c cn1 2 … , to ta-
ku toçku budemo nazyvaty A2 -irracional\nog.
6. Lancghovi A2-droby, wo ne mistqt\ zadanu kombinacig dvox symvo-
liv. Nexaj α1 = 1 2/ , α2 = 1. Poznaçymo çerez C A c c[ ],2 1 2 mnoΩynu lancg-
hovyx A2 -drobiv, wo ne mistqt\ zadanu kombinacig dvox elementiv c c1 2 , c c1 2, ∈
∈ A2 = 1 2 1/ ,{ }. Oçevydno, wo mnoΩyna C A c c[ ],2 1 2 pry c1 ≠ c2 ne mistyt\
A2 -racional\nyx toçok.
Teorema 4. Pry c1 ≠ c2 mnoΩyna C A c c[ ],2 1 2 [ zçyslennog, a pry c1 = c2
� kontynual\nog mnoΩynog, mira Lebeha qko] dorivng[ nulg.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL … 461
Dovedennq. Nexaj c1 ≠ c2 . Todi mnoΩyna C A c c[ ],2 1 2 sklada[t\sq z A2 -
irracional\nyx toçok x , A2 -zobraΩennq qkyx do k - ho miscq vklgçno mistyt\
lyße element c2, k = 1, 2, … , a z ( )k + 1 -ho miscq � lyße element c1, a
takoΩ vklgça[ toçky [( )]1 i [( )]/1 2 . Tomu C A c c[ ],2 1 2 [ zçyslennog.
Nexaj c1 = c2 = c , c̃ � takyj element, wo {˜}c = A c2 \ { }. MnoΩyna
lancghovyx A2 -drobiv z C A cc[ ],2 , wo mistqt\ skinçennu kil\kist\ elementiv
c, [ zçyslennog. Rozhlqnemo toçky x C A cc∈ [ ],2 , v A2 -zobraΩenni qkyx
mistyt\sq neskinçenna kil\kist\ elementiv c. Todi A2 -zobraΩennq takyx toçok
magt\ vyhlqd x =
[ ]˜, ˜, , ˜, , ˜, ˜, , ˜, ,c c c c c c c c
k k
… … …
1 2
1 24 34 1 24 34
. KoΩnomu zobraΩenng
postavymo u vidpovid\ neskinçennu poslidovnist\ natural\nyx çysel k1 , k2 , … .
MnoΩyna takyx poslidovnostej kontynual\na, z çoho vyplyva[ kontynual\-
nist\ mnoΩyny C A cc[ ],2 .
PokaΩemo, wo mira Lebeha mnoΩyny C A cc[ ],2 dorivng[ nulg. Nexaj Fk �
ob�[dnannq cylindriv ranhu k, sered vnutrißnix toçok qkyx [ toçky z mnoΩyny
C A cc[ ],2 . Oçevydno, wo Fk ⊂ Fk−1, k = 1, 2, … , C A cc[ ],2 =
Fkk =
∞
1I i dlq
miry Lebeha mnoΩyny C A cc[ ],2 magt\ misce spivvidnoßennq
λ ( [ ]),C A cc2 = lim ( )
k
kF
→∞
λ ≤ lim max
k
k
c c
c c cn
k
k→∞ …
…
1
1 2
∆ ,
de nk � kil\kist\ cylindriv ranhu k , wo mistqt\sq v Fk . Doslidymo znaçen-
nq nk .
Poznaçymo çerez nk
c
−1
˜ kil\kist\ cylindriv ( )k − 1 -ranhu vyhlqdu ∆c c ck1 2… − ˜ ,
wo vxodqt\ do Fk−1, çerez nk
c
−1 kil\kist\ cylindriv ( )k − 1 -ho ranhu vyhlqdu
∆c c ck1 2… −
, wo vxodqt\ do Fk−1. KoΩnyj cylindr ∆c c ck1 2… − ˜ z Fk−1 porodΩu[
dva cylindry k -ho porqdku: ∆c c cck1 2… − ˜ i ∆c c cck1 2… − ˜ ˜ , qki naleΩat\ Fk , a ko-
Ωen cylindr vyhlqdu ∆c c ck1 2… −
porodΩu[ lyße odyn cylindr ∆c c cck1 2… − ˜ ,
qkyj naleΩyt\ mnoΩyni Fk .
Todi oçevydno, wo n k = 2 1 1n nk
c
k
c
− −+˜ . Oskil\ky nk
c
−1
˜ = nk−2, to nk =
= n n nk
c
k
c
k
c
− − −+ +1 1 1
˜ ˜ = n nk k− −+2 1. Zokrema, n0 = 1, n1 = 2, n2 = 3, i pry n−1 =
= 1 poslidovnist\ { }nk , k = – 1, 0, 1, … , zbiha[t\sq z poslidovnistg çysel
Fibonaççi, z çoho vyplyva[, wo
nk =
5
5
1 5
2
1 5
2
2 2+
− −
+ +k k
.
Na pidstavi naslidku lemy 1 otrymu[mo
max
c c
c c c
k
k
1
1 2…
…∆ =
∆1
2
1
2
1
2
…
k
124 34
≤
1
1
2
1
2
1
21
2
1
qk
k
−
−
…
124 34
=
=
1
2 17
17
1 17
4
1 17
4
2
+
− −
k k
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
462 S. O. DMYTRENKO, D. V. KGRÇEV, M. V. PRAC|OVYTYJ
Todi dlq miry Lebeha mnoΩyny C A cc2,[ ] vykonugt\sq nerivnosti
λ C A cc2,[ ]( ) ≤ lim max
k
k
c c
c c cn
k
k→∞ …
…
1
1 2
∆ ≤
≤ lim
k
k k
k k→∞
+ ++
− −
+
− −
5
5
1 5
2
1 5
2
2 17
17
1 17
4
1 17
4
2 2
2 = 0,
zvidky vyplyva[, wo λ ( [ ]),C A cc2 = 0.
7. Vypadkova velyçyna, elementy lancghovoho A2-zobraΩennq qko] ut-
vorggt\ odnoridnyj lancgh Markova. Nexaj
ξ = [ ], , , ,η η η1 2 … …k
� vypadkova velyçyna (v. v.), zobraΩena lancghovym drobom, elementy ηk qko]
moΩut\ nabuvaty znaçen\ z mnoΩyny A2 = { / },1 2 1 i utvorggt\ odnoridnyj
lancgh Markova { }ηk z poçatkovymy jmovirnostqmy p1
2
> 0 i p1 > 0 i mat-
ryceg perexidnyx imovirnostej pij , i, j A∈ 2 , do toho Ω pij ≥ 0 i p p
i
i1
2
1+ =
= 1, i = 1 2/ , 1. Oçevydno, wo v. v. ξ nabuva[ znaçen\ z mnoΩyny LA2
= [ / ],1 2 1 .
Nahada[mo, wo spektrom rozpodilu v. v. ξ nazyvagt\ mnoΩynu
S = x P x x: ( , ){ }ξ ε ε ε∈ − + > ∀ >{ }0 0 .
MoΩna dovesty, wo spektrom rozpodilu v. v. ξ [ mnoΩyna
A = x x a a a A p kk k a ak k
: , , , , , , ,[ ]= … … ∈ > ∀ = …{ }+1 2 1
0 1 2 .
Teorema 5. Qkwo matrycq perexidnyx imovirnostej pij mistyt\:
1) bil\ße niΩ odyn nul\, to ξ ma[ dyskretnyj rozpodil z dvoma atomamy;
2) til\ky odyn nul\, to ξ ma[:
a) dyskretnyj rozpodil iz zçyslennog mnoΩynog atomiv, qkwo p p1
2
1
2
11 > 0;
b) synhulqrnyj rozpodil kantorivs\koho typu v protyleΩnomu vypadku,
tobto qkwo p p1
2
1 11
2
> 0.
Dovedennq. 1. Matrycq perexidnyx imovirnostej pij ne moΩe maty bil\-
ße niΩ dva nuli. Qkwo pij ma[ dva nuli, to oçevydno, wo rozpodil v. v. ξ [
dyskretnym i atomamy rozpodilu budut\ dvi toçky:
x1 =
1
2
1, ( )
i x2 = [( )]1 pry p1
2
1
2
= p
11
2
= 0,
x1 =
1
2
i x2 = 1
1
2
,
pry p1
2
1
= p11 = 0,
x1 =
1
2
1,
i x2 = 1
1
2
,
pry p1
2
1
2
= p11 = 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 4
LANCGHOVE A2 -ZOBRAÛENNQ DIJSNYX ÇYSEL … 463
x1 =
1
2
i x2 = [( )]1 pry p1
2
1
= p
11
2
= 0.
2. Nexaj teper matrycq pij mistyt\ lyße odyn nul\. MoΩlyvi dva vypad-
ky.
2a. Qkwo p p1
2
1
2
11 > 0, to p1
2
1
= 0 abo p
11
2
= 0. U perßomu vypadku z toç-
nistg do zçyslenno] mnoΩyny rozpodil zoseredΩeno na mnoΩyni C A2
1
2
1,
, u
druhomu � na mnoΩyni C A2 1
1
2
,
. Za teoremog 4 obydvi mnoΩyny [ zçyslen-
nymy, tomu rozpodil v. v. ξ bude dyskretnym iz zçyslennog mnoΩynog atomiv.
2b. Nexaj pcc = 0, de c A∈ 2. Oçevydno, wo spektr rozpodilu v c\omu vy-
padku zbiha[t\sq z mnoΩynog C A cc2,[ ] , dopovnenog A2 -racional\nymy toç-
kamy, wo [ hranyçnymy do ne]. Za teoremog 4 mira Lebeha tako] mnoΩyny doriv-
ng[ nulg. Oskil\ky oçevydno, wo atomiv rozpodilu v c\omu vypadku nema[, to
v. v. ξ ma[ synhulqrnyj rozpodil kantorivs\koho typu [1, c. 69].
Lehko baçyty, wo rozpodil v. v. ξ u vypadku, koly matrycq pij ne mistyt\
nuliv, [ neperervnym. Prote zadaça vyznaçennq typu rozpodilu v c\omu vypadku
zasluhovu[ na okremu uvahu.
1. Prac\ovytyj M. V. Fraktal\nyj pidxid u doslidΩennqx synhulqrnyx rozpodiliv. � Ky]v:
Vyd-vo NPU im. M. P. Drahomanova, 1998. � 296 s.
2. Albeverio S., Baranovskyi O., Pratsiovytyi M., Torbin G. The Ostrogradsky series and related
Cantor-like sets // Acta Arith. – 2007. – 130, # 3. – P. 215 – 230.
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| id | umjimathkievua-article-3033 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:58Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/17/10ed543073edac166bf0c6fd24daa017.pdf |
| spelling | umjimathkievua-article-30332020-03-18T19:43:50Z $A_2$-continued fraction representation of real numbers and its geometry $A_2$-зображення дійсних чисел та його геометрія Dmytrenko, S. O. Kyurchev, D. V. Pratsiovytyi, M. V. Дмитренко, С. O. Кюрчев, Д. В. Працьовитий, М. В. We study the geometry of representations of numbers by continued fractions whose elements belong to the set $A_2 = {α_1, α_2}$ ($A_2$-continued fraction representation). It is shown that, for $α_1 α_2 ≤ 1/2$, every point of a certain segment admits an $A_2$-continued fraction representation. Moreover, for $α_1 α_2 = 1/2$, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose $A_2$-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its $A_2$-continued fraction representation form a homogeneous Markov chain are also investigated. Изучается геометрия представления чисел цепными дробями, элементы которых принадлежат множеству $A_2 = {α_1, α_2}$ (цепное $A_2$-представление). Доказано, что при $α_1 α_2 ≤ 1/2$ каждая точка определенного отрезка имеет цепное A2-представление, причем при $α_1 α_2 = 1/2$ представление единственное, за исключением счетного множества точек. Для последнего случая найдено основное метрическое соотношение, описаны метрические свойства множества чисел, цепное $A_2$-представление которых не содержит заданной комбинации двух элементов, а также изучены свойства случайной величины, элементы цепного $A_2$ -представления которой образуют однородную цепь Маркова. Institute of Mathematics, NAS of Ukraine 2009-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3033 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 4 (2009); 452-463 Український математичний журнал; Том 61 № 4 (2009); 452-463 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3033/2815 https://umj.imath.kiev.ua/index.php/umj/article/view/3033/2816 Copyright (c) 2009 Dmytrenko S. O.; Kyurchev D. V.; Pratsiovytyi M. V. |
| spellingShingle | Dmytrenko, S. O. Kyurchev, D. V. Pratsiovytyi, M. V. Дмитренко, С. O. Кюрчев, Д. В. Працьовитий, М. В. $A_2$-continued fraction representation of real numbers and its geometry |
| title | $A_2$-continued fraction representation of real numbers and its geometry |
| title_alt | $A_2$-зображення дійсних чисел та його геометрія |
| title_full | $A_2$-continued fraction representation of real numbers and its geometry |
| title_fullStr | $A_2$-continued fraction representation of real numbers and its geometry |
| title_full_unstemmed | $A_2$-continued fraction representation of real numbers and its geometry |
| title_short | $A_2$-continued fraction representation of real numbers and its geometry |
| title_sort | $a_2$-continued fraction representation of real numbers and its geometry |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3033 |
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