Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series
We study topological and metric properties of the set $$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$ with certain conditions on the sequence of sets $\{V_n\}$. In particular, we establish con...
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| Дата: | 2007 |
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| Мова: | Українська Англійська |
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Institute of Mathematics, NAS of Ukraine
2007
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509459072155648 |
|---|---|
| author | Baranovskyi, O. M. Pratsiovytyi, M. V. Torbin, H. M. Барановський, О. М. Працьовитий, М. В. Торбін, Г. М. |
| author_facet | Baranovskyi, O. M. Pratsiovytyi, M. V. Torbin, H. M. Барановський, О. М. Працьовитий, М. В. Торбін, Г. М. |
| author_sort | Baranovskyi, O. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:51Z |
| description | We study topological and metric properties of the set
$$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$
with certain conditions on the sequence of sets $\{V_n\}$. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability theory. |
| first_indexed | 2026-03-24T02:41:26Z |
| format | Article |
| fulltext |
UDK 511.72
O. M. Baranovs\kyj (In-t matematyky NAN Ukra]ny, Nac. ped. un-t, Ky]v),
M. V. Prac\ovytyj (Nac. ped. un-t, In-t matematyky NAN Ukra]ny, Ky]v),
H. M. Torbin (Inst. Angewandte Math. Univ., Bonn, Germany, Nac. ped. un-t,
In-t matematyky NAN Ukra]ny, Ky]v
TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN
DIJSNYX ÇYSEL Z UMOVAMY NA }X ROZKLADY
V RQDY OSTROHRADS|KOHO*
We study topological and metric properties of the set
C V x x
g g g g g g
g Vn
n
n
n
k kO1
1
1 1 2 1 2
1
, :
( )
( ) ( )
,{ } = =
−
+ … + + … +
∈ ⊂[ ]
∑
−
N
with certain conditions on the sequence of sets { }Vn . In particular, we establish conditions under which
the Lebesgue measure of this set is: (a) zero; (b) positive. We compare the results obtained with the
corresponding results for continued fractions and discuss their possible applications in the probability
theory.
Yssledugtsq topoloho-metryçeskye svojstva mnoΩestva
C V x x
g g g g g g
g Vn
n
n
n
k kO1
1
1 1 2 1 2
1
, :
( )
( ) ( )
,{ } = =
−
+ … + + … +
∈ ⊂[ ]
∑
−
N
s opredelenn¥my uslovyqmy na posledovatel\nost\ mnoΩestv { }Vn . V çastnosty, ustanovlen¥
uslovyq, pry kotor¥x mera Lebeha πtoho mnoΩestva qvlqetsq: a) nulevoj, b) poloΩytel\noj.
V¥polneno sravnenye s sootvetstvugwymy rezul\tatamy dlq cepn¥x drobej. ObsuΩdagtsq
vozmoΩn¥e prymenenyq poluçenn¥x rezul\tatov v teoryy veroqtnostej.
Vstup. Vidomo bahato riznyx sposobiv podannq ta zobraΩennq dijsnyx çysel
(modelej (interpretacij) aksiomatyçno] teori] dijsnyx çysel) qk za dopomohog
symvoliv skinçennoho, tak i neskinçennoho alfavitiv (naboru cyfr) [1 – 3] . Ko-
Ωen z nyx ma[ svog oblast\ zastosovnosti ta deqki perevahy pered inßymy v qko-
mus\ vidnoßenni. KoΩne zobraΩennq porodΩu[ svo] metryçni spivvidnoßennq i
svog heometrig, na qkyx bazugt\sq metryçna, fraktal\na i jmovirnisna teori]
dijsnyx çysel.
Rozklady çysel v lancghovi droby (lancghove zobraΩennq) zavdqky rozrob-
lenosti teori] ta riznomanitnym zastosuvannqm zajnqly okreme vaΩlyve misce v
matematyci. Razom z cym isnugt\ analohiçni, ale v metryçnomu vidnoßenni vid-
minni, teori], pov’qzani z rozkladamy çysel u znakozminni rqdy. Vony rozrobleni
znaçno menße. Do takyx naleΩat\ zobraΩennq çysel rqdamy Ostrohrads\koho
– Serpins\koho – Pirsa.
Pryblyzno v 1861 r. M.@V.@Ostrohrads\kyj rozhlqnuv dva alhorytmy rozkla-
du dodatnyx dijsnyx çysel u znakozminni rqdy:
k
k
kq q q∑ −
…
−( )1 1
1 2
, N � qk +1 > qk , (1)
k
k
kq∑ − −( )1 1
, qk +1 ≥ q qk k( )+ 1 (2)
(rqdy Ostrohrads\koho 1- i 2-ho vydiv vidpovidno), znajdeni sered rukopysiv i
neopublikovanyx robit c\oho avtora {.@Q.@Remezom [4]. NezaleΩno vid c\oho rq-
* Çastkovo pidtrymano proektamy DFG 436 UKR 113/78, DFG 436 UKR 113/80, SFB-611 ta fon-
dom Aleksandra fon Humbol\dta.
© O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9 1155
1156 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
dy Ostrohrads\koho 1-ho vydu doslidΩuvalys\ inßymy avtoramy. Tak, u roboti
[5] dovedeno, wo suma neskinçennoho rqdu vydu (1) [ irracional\nym çyslom.
U@1911 r. V.@Serpins\kyj [2] rozhlqnuv kil\ka rozvynen\ dijsnyx çysel v rqdy,
sered qkyx i rozvynennq (1) ta (2). Vin zaznaça[, wo rozvynennq vydu (1) zustri-
çagt\sq u knyzi G.@Puzyny „Teorya funkcyi analitycznych” (1898 r.). U 1929 roci
rozklady çysel v rqdy vydu (1) z’qvlqgt\sq u roboti T.@Pirsa [6] bez bud\-qkyx
posylan\ na inßi roboty. V anhlomovnij literaturi taki rozklady nazyvagt\
rozkladamy Pirsa, wo [ ne zovsim vypravdanym. Zaznaçymo, wo v roboti [7]
moΩna znajty dodatkovu informacig pro zarodΩennq idej ta rozvytok teori]
rqdiv Ostrohrads\koho – Serpins\koho – Pirsa. My, dotrymugçys\ vitçyznqno]
tradyci] [4, 8 – 12], nazyva[mo rozklady çysel v rqdy vydu (1) rqdamy Ostro-
hrads\koho 1-ho vydu. We kil\ka robit prysvqçeno doslidΩenng riznyx pytan\,
pov’qzanyx z rozkladamy dijsnyx çysel u rqdy Ostrohrads\koho, zokrema vy-
vçenng pytan\ podannq çysel pevnoho vydu rqdamy vyhlqdu (1) [13 – 17], roz-
v’qzanng deqkyx metryçnyx zadaç [18, 19, 7, 20] ta in. [21 – 23, 7, 20]. Krim
roboty [13] avtoram dano] statti buly nedostupni inßi roboty DΩ.@Íallita, xo-
ça vidomo pro ]x isnuvannq [18, 22].
KoΩne dijsne çyslo x ∈( , )0 1 moΩna podaty u vyhlqdi skinçennoho çy ne-
skinçennoho rqdu (1). Qkwo çyslo x [ irracional\nym, to ce moΩna zrobyty
[dynym çynom, i vyraz (1) pry c\omu [ neskinçennym; qkwo Ω x [ irracional\-
nym, to joho moΩna podaty u vyhlqdi (1) zi skinçennog kil\kistg dodankiv dvoma
riznymy sposobamy [4].
Rqdy Ostrohrads\koho zbihagt\sq dosyt\ ßvydko, najpovil\niße zbiha[t\sq
rqd
k
k
k
e
e=
∞ −
∑ − = −
1
11 1( )
!
.
Ce dozvolq[ nablyΩaty irracional\ni çysla çyslamy racional\nymy, wo [ çast-
kovymy sumamy rqdu Ostrohrads\koho. Pry c\omu poxybka ne perevywu[ modu-
lq perßoho z vidkynutyx dodankiv:
x
q x q x q xk
m k
k
− −
…=
−
∑
1
1
1 2
1( )
( ) ( ) ( )
=
=
j
j
m jq x q x q x=
∞ −
+
∑ −
…1
1
1 2
1( )
( ) ( ) ( )
< 1
1 2 1q x q x q xm( ) ( ) ( )… +
.
Qkwo poklasty
g q1 1= ,
g q q2 2 1= − ,
………………,
g q qn n n+ += −1 1 ,
……………… ,
to vyraz (1) moΩna perepysaty u vyhlqdi
1 1 1
1 1 1 2
1
1 1 2 1 2g g g g g g g g g g
n
n
−
+
+ … + −
+ … + + … +
+ …
−
( )
( )
( ) ( )
. (3)
Vyraz (3) skoroçeno zapysu[t\sq u vyhlqdi
O1
1 2( , , , , )g g gn… …
i nazyva[t\sq O1
-zobraΩennqm çysla x ∈( , )0 1 . Çysla gn nazyvagt\ O1-
symvolamy çysla x.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN DIJSNYX ÇYSEL Z UMOVAMY 1157
My doslidΩu[mo metryçni vlastyvosti mnoΩyny C VnO1, { }[ ] vsix çysel
x ∈[ ]0 1, , O1
-symvoly qkyx zadovol\nqgt\ umovu g xn( ) ∈ @Vn ⊂ @N dlq vsix n .
Zadaça znaxodΩennq miry Lebeha mnoΩyny C VnO1, { }[ ] ma[ samostijnyj inte-
res i cikava takoΩ u zv’qzku z doslidΩennqm rozpodiliv vypadkovyx velyçyn
vyhlqdu
ξ = O1
1 2ξ ξ ξ, , , ,… …( )k ,
de ξk — vypadkovi velyçyny. U cij roboti znajdeno umovy dodatnosti miry
Lebeha mnoΩyny C VnO1, { }[ ] u vypadkax, koly mnoΩyny N \ Vk [ skinçennymy
i koly mnoΩyny Vk [ skinçennymy, a takoΩ znajdeno umovy rivnosti 0 miry
Lebeha mnoΩyny C VnO1, { }[ ] u vypadkax, koly mnoΩyny Vk [ skinçennymy i
koly mnoΩyny Vk ta N \ Vk [ neskinçennymy. Cq stattq prodovΩu[ doslid-
Ωennq, rozpoçati v robotax [24 – 27].
1. Cylindryçni mnoΩyny ta ]x vlastyvosti. Cylindryçnog mnoΩynog
(cylindrom) ranhu m z osnovog c c cm1 2, , ,…( ) nazyvagt\ mnoΩynu O[ ]c c cm1 2
1
…
vsix çysel x ∈[ ]0 1, , qki moΩna podaty u vyhlqdi (3) tak, wo m perßyx O1
-
symvoliv çysla x dorivnggt\ c1, c2, … , cm vidpovidno.
Lehko dovesty, wo cylindryçni mnoΩyny magt\ taki vlastyvosti.
1. Cylindryçna mnoΩyna O[ ]c c cm1 2
1
… [ vidrizkom [ , ]a b , de
a = min ( , , , ), ( , , , )O O1
1 2
1
1 2 1c c c c c cm m… … +{ },
b = max ( , , , ), ( , , , )O O1
1 2
1
1 2 1c c c c c cm m… … +{ }.
ZauvaΩennq 1. Interval z tymy Ω kincqmy, wo j vidrizok O[ ]c c cm1 2
1
… , bu-
demo poznaçaty O( )c c cm1 2
1
… i nazyvaty cylindryçnym intervalom.
2. O[ ]c c cm1 2
1
… =
c c c c c mm
c c c=
∞
… …
1
1 1
1 21 2∪ ∪O O[ ] ( , , , ), pryçomu
sup inf[ ] [ ( )]O Oc c c c c c c cm m1 2 1 2
1
1
1
… … += , qkwo m [ neparnym,
inf sup[ ] [ ( )]O Oc c c c c c c cm m1 2 1 2
1
1
1
… … += , qkwo m [ parnym,
i
O O O[ ] [ ( )] ( , , , , )c c c c c c c c mm m
c c c c
1 2 1 2
1
1
1 1
1 2 1… … + = … +∩ .
3. DovΩyna cylindryçno] mnoΩyny O[ ]c c cm1 2
1
… vyznaça[t\sq rivnistg
O c c c
m m
m[ ] ( )1 2
1
1 2
1
1… =
… +σ σ σ σ
,
de σk =
i
k
ic=∑ 1
.
Lema 1 [27]. Vidnoßennq dovΩyn cylindryçnyx mnoΩyn O[ ]c c c sm1 2
1
… t a
O[ ]c c cm1 2
1
… zadovol\nq[ rivnist\
O
O
[ ]
[ ]
( )( )
( )
c c c s
c c c
s
m
m
a
a s a s
f a1 2
1 2
1
1 1
…
…
=
+ − +
= , (4)
de a = 1 +
i
m
ic=∑ 1
. Krim toho,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1158 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
f as( ) ≤ 1
2 2 1⋅ −( )s
(5)
i dlq m ≥ s – 1
O
O
[ ]
[ ]
( )( )
c c c s
c c c
m
m
m
m s m s
1 2
1 2
1
1
1
1
…
…
≤ +
+ + +
. (6)
ZauvaΩennq 2. Nexaj
∆c c cm1 2 …
l d. .
— cylindryçna mnoΩyna, porodΩena zo-
braΩennqm çysel lancghovymy drobamy. Vidomo [28, s.@75], wo
∆
∆
c c c s
c c c
m
m
m
m
m
m
m
m
s
Q
Q
Q
sQ s
Q
sQ
1 2
1 2
1
1
1 1 12
1
1 1
…
…
−
− −
=
+
+
+ +
l d
l d
. .
. .
,
de Qk poznaça[ znamennyk pidxidnoho drobu porqdku k lancghovoho drobu
c c cn1 2, , , ,… …[ ]
i vyznaça[t\sq rekurentnog rivnistg
Q0 1= , Q c1 1= , Q c Q Qk k k k= +− −1 2 dlq bud\-qkoho k ≥ 2.
Podvijna nerivnist\
1
3 2s
<
∆
∆
c c c s
c c c
m
m
1 2
1 2
…
…
l d
l d
. .
. .
< 2
2s
ma[ misce dlq bud\-qkyx natural\nyx c1, c2, … , cm i s. Dlq O1
-zobraΩennq
ma[mo f as( ) → 0, a → ∞ , i lema 1 svidçyt\ pro pryncypovu vidminnist\ osnov-
nyx metryçnyx spivvidnoßen\ zobraΩennq çysel rqdamy Ostrohrads\koho 1-ho
vydu ( O1
-zobraΩennq) i zobraΩennq çysel lancghovymy drobamy.
2. Topoloho-metryçni vlastyvosti mnoΩyny C VnO1, {{ }}[[ ]]. Nexaj { }Vn
— fiksovana poslidovnist\ neporoΩnix pidmnoΩyn mnoΩyny N natural\nyx
çysel. Rozhlqnemo mnoΩynu C VnO1, { }[ ], qka [ zamykannqm mnoΩyny
C Vn
* ,O1 { }[ ] usix irracional\nyx çysel x = O1
1(g x( ), g x2( ), … , g xn( ), … ), O1
-
symvoly qkyx zadovol\nqgt\ umovu
g x Vn n( ) ∈ (7)
dlq vsix n ∈ N.
ZauvaΩennq 3. MnoΩyna C V C Vn nO O1 1, \ ,*{ }[ ] { }[ ] [ pidmnoΩynog mno-
Ωyny racional\nyx çysel, a tomu mnoΩyny C VnO1, { }[ ] i C Vn
* ,O1 { }[ ] magt\
odnakovu miru Lebeha λ.
Oçevydno, wo:
1) C VnO1, { }[ ] = [ , ]0 1 , qkwo Vn = N dlq vsix n ∈ N;
2) mnoΩyna C VnO1, { }[ ] [ ob’[dnannqm vidrizkiv i, otΩe, ma[ dodatnu miru
Lebeha, qkwo Vn = N dlq vsix n > n0 .
Tomu my rozhlqdatymemo lyße vypadok, koly Vn ≠ N dlq neskinçenno]
mnoΩyny znaçen\ n.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN DIJSNYX ÇYSEL Z UMOVAMY 1159
Vvedemo poznaçennq, qki budemo vykorystovuvaty pry doslidΩenni mnoΩyny
C VnO1, { }[ ]. Zapys A −~ B oznaça[, wo symetryçna riznycq mnoΩyn A i B
(mnoΩyna A B\( ) ∪ B A\( )) [ ne bil\ß niΩ zçyslennog, tobto mnoΩyny A i B
zbihagt\sq z toçnistg do zçyslenno] mnoΩyny.
Poznaçymo çerez Fk ob’[dnannq vsix cylindriv ranhu k, vnutrißnist\ qkyx
mistyt\ toçky mnoΩyny C VnO1, { }[ ], F0 = [ , ]0 1 , a Fk +1 oznaçymo rivnistg
Fk +1 = F Fk k\ +1.
Todi
Fk −~
c V c V
c c c
k k
k
1 1
1 2
1
∈ ∈
……∪ ∪ O[ ],
Fk +1 −~
c V c V s V
c c c s
k k k
k
1 1 1
1 2
1
∈ ∈ ∉
……
+
∪ ∪ ∪ O( ) ,
F F F F F F Fk k k k k k
i
k
i= ⇔ = =+ + + +
=
+
1 1 1 1
1
1
0 1∪ ∪\ [ , ] \ ,
i mira Lebeha mnoΩyn Fk i Fk +1 vyznaça[t\sq rivnostqmy
λ
σ σ σ σ
( )
( )
Fk
c V c V k kk k
= …
… +∈ ∈
∑ ∑
1 1
1
11 2
,
λ( )Fk +1 =
c V c V s V k k kk k k
s s
1 1 1
1
11 2∈ ∈ ∉
∑ ∑ ∑…
… + + +
+
σ σ σ σ σ( )( )
=
=
c V c V k s V k kk k k
s s
1 1 1
1 1
11 2∈ ∈ ∉
∑ ∑ ∑…
… + + +
+
σ σ σ σ σ( )( )
.
Z oznaçen\ C VnO1, { }[ ], Fk , Fk +1 i neperervnosti miry Lebeha λ( )⋅ vyply-
va[ nastupne tverdΩennq.
Lema 2. Magt\ misce vidnoßennq
C VnO1, { }[ ] ⊂ Fk
c
+1 ⊂ Fk
c, C VnO1, { }[ ] =
k
k
cF
=
∞
1
∩
i
C VnO1, { }[ ] −~ [ , ] \0 1
1k
kF
=
∞
∪ ,
de Ec
— zamykannq mnoΩyny E, a otΩe,
λ
σ σ σ σ
λC V Fn
c V c V k k
k
k k
O1
1 21 1
1
1
,
( )
( ){ }[ ]( ) ≤ …
… +
=
∈ ∈
∑ ∑ ,
λ λC V Fn
k
kO1, lim ( ){ }[ ]( ) =
→∞
, (8)
λ λC V Fn
k
kO1
0
11, { }[ ]( ) = − ( )
=
∞
+∑ .
Lema 3 [25]. MnoΩyna C VnO1, { }[ ] [ doskonalog mnoΩynog (tobto zamk-
nenog mnoΩynog bez izol\ovanyx toçok). Qkwo Vn ≠ N dlq neskinçenno] mno-
Ωyny znaçen\ n, to vona [ nide ne wil\nog mnoΩynog.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1160 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
Lema 4. Qkwo λ C VnO1, { }[ ]( ) > 0, to
lim lim
k
k
k k
k
k
F
F
F
F→∞
+
→∞
+
+
( )
( )
=
( )
( ) =
λ
λ
λ
λ
1 1
1
0 .
Dovedennq. Oskil\ky ma[ misce rivnist\ (8), to tverdΩennq lemy vyplyva[ z
rivnosti
λ Fk +( )1 = λ Fk( ) @–@ λ Fk +( )1 .
Naslidok 1. Qkwo
lim
k
k
k
F
F→∞
+( )
( )
≠
λ
λ
1 0 ,
to λ C O Vn
1, { }[ ]( ) = 0.
Lema 5 [27]. MnoΩyna C VnO1, { }[ ] ma[ nul\ovu miru Lebeha todi i til\ky
todi, koly
k
k
k
F
F=
∞
+∑ ( )
( )
= ∞
1
1λ
λ
. (9)
Naslidok 2. Qkwo dlq deqko] dodatno] konstanty c
λ
λ
F
F
k
k
+( )
( )
1 ≥ c > 0,
to λ C VnO1, { }[ ]( ) = 0.
3. Umovy dodatnosti miry Lebeha mnoΩyny C VnO1, {{ }}[[ ]]. Zafiksu[mo n
∈ ∈ N i nabir natural\nyx çysel (s1, s2 , … , sn), de s V1 1∈ , s V2 2∈ , … , s Vn n∈ .
Vvedemo poznaçennq
F Fk
s s s
n k s s s
n
n
1 2
1 2
1…
+ …= ∩ O[ ],
F Fk
s s s
n k s s s
n
n+
…
+ + …=1 1
11 2
1 2
∩ O[ ].
Spoçatku rozhlqnemo vypadok, koly
V v v vk k k k= { … } = { + + …}N \ , , , , ,1 2 1 2 , vk ∈N .
Lema 6. Nexaj Vk = { vk + 1, vk + 2, … }, vk ∈N . Todi
λ Fk
s s sn
+
…( )1
1 2 < 1
2
1 1 2v
v
Fn k
n k
k
s s sn+ +
+
…( )λ (10)
dlq bud\-qkoho natural\noho k.
Dovedennq. Viz\memo dovil\nyj fiksovanyj cylindryçnyj interval
O( )c c cn k1 2
1
… +
ranhu n + k. Todi
c V
c c c
n k n k
n k
+ +
+
∉
…∑ O( )1 2
1 =
=
c
v
n k n k n k n k n kn k
n k
c c
+
+
= + − + − + + − +
∑ … + + +1 1 2 1 1 1
1
1σ σ σ σ σ( )( )
>
>
v
v v
n k
n k n k n k n k n k
+
+ − + − + + − +… + + + +σ σ σ σ σ1 2 1 1 11 2( )( )
.
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TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN DIJSNYX ÇYSEL Z UMOVAMY 1161
Teper viz\memo dovil\nyj fiksovanyj cylindryçnyj interval O c c c cn k( )1 2
1
… +
ran-
hu n + k + 1. Todi
c V c V
c c c c
n k n k n k
n k
+ + + +
+
∈ ∉
…∑ ∑
1
1 2
1O( ) =
=
c v c
v
n k n k n k n kn k n k
n k
c c
+ +
+ +
= +
∞
= + + + − +
∑ ∑ … + + +1 1 1 2 1
1
1
1σ σ σ σ σ σ( )( )
≤
≤
c v
n k
n k n k n k n kn k n k
v
+ += +
∞
+ +
+ − + + +
∑ … + +1
1
1 2 1 1 2σ σ σ σ σ σ( )( )
=
= 1
2 1 2
1
1 2 1 1 1
v
v v
n k
n k n k n k n k n k
+ +
+ − + − + + − +… + + + +σ σ σ σ σ( )( )
.
OtΩe,
c V c V
c c c c
n k n k n k
n k
O
+ + + +
+
∈ ∉
…∑ ∑
1
1 2
1
( ) < 1
2
1 1
1 2
v
v
n k
n k c V
c c c
n k n k
n k
+ +
+ ∉
…
+ +
+∑ O( ) .
Zafiksuvavßy c1 = s1 ∈ V1, c2 = s2 ∈ V2, … , cn = sn ∈ Vn ta pidsumuvavßy
obydvi çastyny ostann\o] nerivnosti po vsix cn +1 ∈ Vn +1, cn + 2 ∈ Vn + 2, …
… , cn k+ −1 ∈ Vn k+ −1, otryma[mo nerivnist\ (10).
Teorema 1. Nexaj Vk = { vk + 1, vk + 2, … }, vk ∈N ,
k
k
k
v
=
∞
∑
1 2
< + ∞
i Sn =
k n
k
k
v
=
∞∑
2
. Qkwo 4 3S > v1, to
λ C VnO1, { }[ ]( ) > 1
4
1
13
2
2S
v
v
−
+
> 0.
Qkwo Ω 4 3S ≤ v1, to
λ C VnO1, { }[ ]( ) > 1
1
1
1
4
11
2
2
2
1 2v
v
v
S
v v+
−
+ + +
> 0.
Dovedennq. Nexaj O[ ]c1
1
— deqka cylindryçna mnoΩyna z c1 > v1 i
∆c n cC V
1 1
1 1= { }[ ]O O, [ ]∩ .
Todi
λ F c
1
1( ) =
s
v
c c s c s=
∑ + + +1 1 1 1
2 1
1( )( )
=
v
c c c v
2
1 1 1 21 1( )( )+ + +
=
v
c v c
2
1 2
1
1 1+ +
O[ ]
i zrozumilo, wo
λ( )∆c1
= O[ ]c1
1 –
k
k
cF
=
∞
∑ ( )
1
1λ .
Z lemy 6 vyplyva[, wo
λ Fk
c1( ) <
v
v
Fk
k
c+
− ( )1
1
2
12
1λ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1162 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
OtΩe,
λ( )∆c1
= O[ ]c1
1 –
k
k
k
cv
v
F
=
∞
+
−∑ ( )
1
1
1
2
12
1λ =
= O[ ]c1
1 –
k
k
k c
v
v
v
c v=
∞
+
−∑ + +1
1
1
2
2
1 2
1
2 1 1
O[ ] =
= O[ ]c1
1 1 4
1 21 2 1
1
1−
+ +
=
∞
+
+∑c v
v
k
k
k = O[ ]c1
1 1
4
1
2
1 2
−
+ +
S
c v
.
Nerivnist\
1
4
1
2
1 2
−
+ +
S
c v
> 0
rivnosyl\na nerivnosti
c1 > 4 3S – 1.
OtΩe, pry umovi 4 3S > v1 dlq dovil\noho c1 ≥ 4 3S ma[ misce λ( )∆c1
> 0. Todi
λ C VnO1, { }[ ]( ) =
c v
c
1 1
1
1= +
∞
∑ λ( )∆ >
c S
c
1 3
1
4
1
=
∞
∑ O[ ] 1
4
1
2
1 2
−
+ +
S
c v
=
=
c S c c
1 34 1 1
1
1=
∞
∑ +( )
– 4 2S
c S c c c v
1 34 1 1 1 2
1
1 1=
∞
∑ + + +( )( )
=
= 1
4 3S
– 4 1
1
1
1 13
4 1 1 2 1 1 21 2
S
c c v c c vc S=
∞
∑ + +
−
+ + +
( ) ( )( )
.
Ostanng sumu moΩna zapysaty u vyhlqdi
c S v c c v
1 34 2 1 1 2
1
1
1 1
1=
∞
∑ +
−
+ +
–
c S v c c v
1 34 2 1 1 2
1 1
1
1
1=
∞
∑ +
−
+ +
=
= 1
1
1
4
1
42 3 3 2v S S v+
+ … +
+
– 1 1
4 1
1
42 3 3 2v S S v+
+ … +
+
=
= 1
4 13 2S v( )+
– 1
1
1
4 1
1
42 2 3 3 2v v S S v( )+ +
+ … +
+
≥
≥ 1
4 13 2S v( )+
– 1
12 2v v( )+
v
S v
2
3 24 +
,
tomu
λ C VnO1, { }[ ]( ) > 1
4
4
1
1
4
1
43
2
2 3 3 2S
S
v S S v
−
+
−
+
=
= 1
4
1
1
4
43
2
2
2
3 2S
v
v
S
S v
−
+ +
= 1
4
1
13
2
2S
v
v
−
+
,
oskil\ky 4 2S = 4 3S + v2 .
Qkwo 4 3S ≤ v1, to
1
4
1
02
1 2
−
+ +
>S
c v
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN DIJSNYX ÇYSEL Z UMOVAMY 1163
dlq dovil\noho c1 > v1, tomu, zaminyvßy u poperednix mirkuvannqx 4 3S na
v1 + 1, otryma[mo
λ C VnO1, { }[ ]( ) > 1
1
1
1
4
11
2
2
2
1 2v
v
v
S
v v+
−
+ + +
.
Naslidok 3. Qkwo lim
k
k
k
v
v→∞
+1 < 2, to λ C VnO1, { }[ ]( ) > 0.
Naslidok 4 [27]. Qkwo vk = m dlq dovil\noho k ∈N , to
λ C VnO1, { }[ ]( ) > 1
1 2( )m +
.
ZauvaΩennq 4. Porivnq[mo teoremu 1 z vidpovidnym tverdΩennqm teori]
lancghovyx drobiv (LD). Nexaj
C VnLD, { }[ ] — mnoΩyna, analohiçna mnoΩyni
C VnO1, { }[ ], tobto vona [ zamykannqm mnoΩyny vsix irracional\nyx çysel
x = a x a x a xn1 2( ), ( ), , ( ),… …[ ],
elementy a xn( ) lancghovoho drobu qkyx zadovol\nqgt\ umovu
a x Vn n( ) ∈
dlq vsix n ∈N (qk i raniße, { }Vn — fiksovana poslidovnist\ neporoΩnix pid-
mnoΩyn mnoΩyny N natural\nyx çysel). Qkwo, napryklad, Vn = V = N \ { }1
dlq vsix n ∈N , to mira Lebeha mnoΩyny C VnLD, { }[ ] dorivng[ 0 (dyv. [3], te-
oremu 4.2.3). Ale vidpovidna mnoΩyna C VnO1, { }[ ] ma[ dodatnu miru Lebeha.
Takym çynom, teorema 1 svidçyt\ pro we odnu pryncypovu vidminnist\ metryçno]
teori] rqdiv Ostrohrads\koho ta metryçno] teori] lancghovyx drobiv.
Rozhlqnemo inßyj klas mnoΩyn C VnO1, { }[ ], a same, nexaj
V mk k= …{ }1 2, , , , mk ∈N .
Teorema 2. Qkwo Vk = {1, 2, … , mk }, mk ∈N , pryçomu
k
k
k
m m m
m=
∞
+
∑ + + … + < ∞
1
1 2
1
,
to mira Lebeha λ C VnO1, { }[ ]( ) > 0.
Dovedennq. Oskil\ky σk ≤ m1 + m2 + … + mk i
σ
σ
k
k k
k
k km
m m
m m m
+
+ +
≤ + … + +
+ … + + ++ +
1
1
1
11
1
1 1
,
to
λ Fk +( )1 =
c V c V k k
k
k kk k
m
1 1
1
1
1
11 2 1∈ ∈ +
∑ ∑…
… +
+
+ +
σ σ σ σ
σ
σ( )
≤
≤
m m
m m m
Fk
k k
k
1
1 1
1
1
+ … + +
+ … + + ++
λ( ),
tobto
λ
λ
F
F
m m
m m m
k
k
k
k k
+
+
( )
≤ + … + +
+ … + + +
1 1
1 1
1
1( )
.
Rqdy
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1164 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
k
k
k k
m m
m m m=
∞
+
∑ + … + +
+ … + + +1
1
1 1
1
1
i
k
k
k
m m
m=
∞
+
∑ + … +
1
1
1
zbihagt\sq abo rozbihagt\sq odnoçasno. Tomu zi zbiΩnosti ostann\oho vyplyva[
zbiΩnist\ rqdu
k
k
k
F
F=
∞
+∑ ( )
1
1λ
λ( )
,
wo zhidno z lemog 5 [ rivnosyl\nym dodatnosti miry Lebeha mnoΩyny
C VnO1, { }[ ].
Teoremu 2 dovedeno.
4. Umovy nul\mirnosti mnoΩyny C VnO1, {{ }}[[ ]].
Teorema 3. Qkwo Vk = {1, 2, … , mk }, mk ∈N , pryçomu
k k
k
m=
∞
∑
1
= ∞,
to λ C VnO1, { }[ ]( ) = 0.
Dovedennq. Rozhlqnemo
λ Fk +( )1 =
c V c V k s m k kk k k
s s
1 1 1
1 1
11 2 1∈ ∈ = +
∞
∑ ∑ ∑…
… + + +
+
σ σ σ σ σ( )( )
=
=
c V c V k k kk k
m
1 1
1
11 2 1∈ ∈ +
∑ ∑…
… + +σ σ σ σ( )
=
=
c V c V k k
k
k kk k
m
1 1
1
1
1
11 2 1∈ ∈ +
∑ ∑…
… +
+
+ +
σ σ σ σ
σ
σ( )
.
Ocinymo vidnoßennq
σ
σ
k
k km
+
+ ++
1
11
.
Oskil\ky k < σk + 1 i
a
a b
a c
a c b+
< +
+ +( )
dlq dovil\nyx a, b, c ∈ N,
to
k
k mk+ +1
<
σ
σ
k
k km
+
+ ++
1
11
.
Todi
λ Fk +( )1 > k
k mk c V c V k kk k
+
…
… ++ ∈ ∈
∑ ∑
1 1 21 1
1
1σ σ σ σ( )
= k
k mk+ +1
λ( )Fk ,
tobto
k
k mk+ +1
<
λ
λ
F
F
k
k
+( )1
( )
i
k k
k
k m=
∞
+
∑ +1 1
<
k
k
k
F
F=
∞
+∑ ( )
1
1λ
λ( )
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN DIJSNYX ÇYSEL Z UMOVAMY 1165
Vraxovugçy, wo rqdy
k k
k
k m=
∞
+
∑ +1 1
,
k k
k
m=
∞
+
∑
1 1
,
k k
k
m=
∞
∑
1
odnoçasno zbihagt\sq abo rozbihagt\sq, ma[mo
k k
k
m=
∞
∑
1
= ∞ ⇒
k
k
k
F
F=
∞
+∑ ( )
1
1λ
λ( )
= ∞,
wo zhidno z lemog 5 [ rivnosyl\nym rivnosti λ C O Vn
1, { }[ ]( ) = 0.
Teoremu 3 dovedeno.
Naslidok 5 [27]. Qkwo Vk = {1, 2, … , mk } i rqd
k km=
∞
∑
1
1
rozbiha[t\sq, to λ C VnO1, { }[ ]( ) = 0.
Naslidok 6 [27]. Qkwo Vk = {1, 2, … , m} dlq vsix natural\nyx k, to
λ C VO1, { }[ ]( ) = 0.
I, nareßti, rozhlqnemo vypadok, koly i mnoΩyny Vk , i mnoΩyny N \ Vk [ ne-
skinçennymy.
Teorema 4. Qkwo Vk = N \ , , , ,( ) ( ) ( )a a ak k
n
k
1 2 … …{ }, de an
k
+1
( ) – an
k( ) ≤ d k( ) , 2 ≤
≤ d k( ) ∈ N, k = 1, 2, … , pryçomu
k
k kd a=
∞
∑
1 1
1
( ) ( ) = ∞, (11)
to mira Lebeha mnoΩyny C VnO1, { }[ ] dorivng[ 0.
Dovedennq. Nexaj O[ ]c c ck1 2 1
1
… −
— dovil\na cylindryçna mnoΩyna taka, wo
c Vi i∈ . Rozhlqnemo
c V
c c c c
k
k
∉
…∑ −
O[ ]1 2 1
1 = 1 1
11 2 1 1 1 1σ σ σ σ σ… +( ) + +( )− =
∞
− −
∑
k n k n
k
k n
ka a( ) ( ) .
Nexaj a k
1
( ) = a k
1
( )
, an
k
+1
( ) = an
k( ) + d k( )
. Todi an
k( ) ≥ an
k( )
i
c V
c c c c
k
k
∉
…∑ −
O[ ]1 2 1
1 > 1 1
1 2 1 1 1 1σ σ σ σ σ… +( ) + +( )− =
∞
− −
∑
k n k n
k
k n
k ka a d( ) ( ) ( ) =
= 1 1
1 2 1 1 1σ σ σ σ… +( )− −k
k
k
kd a( ) ( ) .
Vraxovugçy, wo
1 1
11 1 1 1σ σk
k k
ka a− −+
≥
+( ) ( )( )
,
ma[mo
c V
c c c c
k
k
∉
…∑ −
O[ ]1 2 1
1 > 1
1d ak k( ) ( ) O[ ]c c ck1 2 1
1
… −
.
Todi
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1166 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
λ( )Fk =
c V c V c V
c c c c
k k k
k
1 1 1 1
1 2 1
1
∈ ∈ ∉
…∑ ∑ ∑…
− −
−
O[ ] >
> 1
1d ak k( ) ( )
c V c V
c c c
k k
k
1 1 1 1
1 2 1
1
∈ ∈
…∑ ∑…
− −
−
O[ ] = 1
1d ak k( ) ( ) λ( )Fk−1 ,
tobto
λ( )Fk > 1
1d ak k( ) ( ) λ( )Fk−1 .
Tomu
λ( )Fk = λ( )Fk−1 – λ( )Fk < 1 1
1
−
d ak k( ) ( ) λ( )Fk−1
i
λ( )Fk < λ( ) ( ) ( )F
d ai
k
i i1
1
1
1
1 1
=
−
∏ −
.
OtΩe,
λ C VnO1, { }[ ]( ) = lim ( )
k
kF
→∞
λ ≤ λ( ) ( ) ( )F
d ai
i i1
1 1
1 1
=
∞
∏ −
.
Pry vykonanni umovy (11) ostannij neskinçennyj dobutok rozbiha[t\sq do 0, a
otΩe, λ C VnO1, { }[ ]( ) = 0, wo j potribno bulo dovesty.
Naslidok 7 [27]. Qkwo dlq vsix natural\nyx k
Vk = V = N \ , , , ,a a an1 2 … …{ },
de an +1 – an < d, 2 ≤ d — fiksovane natural\ne çyslo, to λ C VnO1, { }[ ]( ) = 0.
Naslidok 8 [27]. Qkwo Vk = V = { v1, v2 , … , vn , … } dlq vsix natural\nyx
k, pryçomu vn +1 – vn ≥ 2 dlq koΩnoho n, bil\ßoho deqkoho fiksovanoho n0 ,
to λ C VnO1, { }[ ]( ) = 0.
Naslidok 9. Qkwo Vk = v v vk k
n
k
1 2
( ) ( ) ( ), , , ,… …{ }, pryçomu vn
k
+1
( ) – vn
k( )
≥ 2
dlq koΩnoho n > nk , k = 1, 2, … , i
k n
kv
k=
∞
∑
1
1
( ) = ∞,
to λ C VnO1, { }[ ]( ) = 0.
5. Vykorystannq otrymanyx rezul\tativ dlq doslidΩennq rozpodiliv
imovirnostej. Rozhlqnemo vypadkovu velyçynu
ξ η η η= … …O1( , , , , )1 2 k ,
O1
-symvoly qko] [ nezaleΩnymy, nabuvagt\ znaçen\ 1, 2, … , i, … z imovirnos-
tqmy p k1 , p k2 , … , pik , … vidpovidno ( pik ≥ 0, p k1 + … + pik + … = 1). Vona
vyvçalasq u robotax [25, 27]. Oskil\ky spektr Sξ vypadkovo] velyçyny ξ [ za-
mykannqm mnoΩyny
x x g x g x g x pk g x kk
: ( ), ( ), , ( ), , ( )= … …( ) >{ }O1
1 2 0 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
TOPOLOHO-METRYÇNI VLASTYVOSTI MNOÛYN DIJSNYX ÇYSEL Z UMOVAMY 1167
tobto z toçnistg do zçyslenno] mnoΩyny [ mnoΩynog typu C VnO1, { }[ ], to
otrymani v poperednix punktax rezul\taty dozvolqgt\ vkazaty dostatni umovy
naleΩnosti rozpodilu do riznyx çystyx typiv synhulqrnyx rozpodiliv (C-typ
abo kantorivs\kyj, S-typ abo salemivs\kyj, P-typ abo kvazikantorivs\kyj [3]).
Budemo vvaΩaty, wo umova neperervnosti rozpodilu ξ [25] vykonu[t\sq:
k i
ikp
=
∞
∏ { } =
1
0max .
Z teorem 3 ta 4 vyplyva[ nastupne tverdΩennq.
Teorema 5. Qkwo „matrycq” pik zadovol\nq[ umovy
pik > 0 , i mk∈{ … }1 2, , , ,
pjk = 0 , j mk> , k ∈N ,
k k
k
m=
∞
∑ = ∞
1
abo
pjk = 0, j W a a ak
k k
n
k∈ = … …{ }1 2
( ) ( ) ( ), , , , , a a dn
k
n
k k
+ − ≤1
( ) ( ) ( )
, 2 ≤ ∈d k( )
N ,
pik > 0 , i V Wk k∈ = N \ , k ∈N ,
k
k k
k
d a=
∞
∑ = ∞
1 1
( ) ( ) ,
to rozpodil ξ [ synhulqrnym rozpodilom kantorivs\koho typu λ ξ( )S =( )0 .
Vykorystovugçy teoremy 1 ta 2, lehko dovesty nastupne tverdΩennq.
Teorema 6. Qkwo „matrycq” pik zadovol\nq[ umovy
pjk > 0 , j vk∈{ … }1 2, , , ,
pik > 0 , j vk> ,
k
k
kv
=
∞
−∑ < ∞
1
2 ,
abo
pik > 0 , i mk∈{ … }1 2, , , ,
pjk = 0 , j mk> ,
k
k
k
m m m
m=
∞
+
∑ + + … + < ∞
1
1 2
1
,
to spektr Sξ rozpodilu ξ [ nide ne wil\nog mnoΩynog dodatno] miry Lebeha.
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ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
1168 O. M. BARANOVS|KYJ, M. V. PRAC|OVYTYJ, H. M. TORBIN
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# 3. – P. 198 – 202.
18. Shallit J. O. Metric theory of Pierce expansions // Ibid. – 1986. – 24, # 1. – P. 22 – 40.
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OderΩano 18.08.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 9
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| id | umjimathkievua-article-3379 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:26Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5e/f4c28f6c02a71730670c1a57d9d9545e.pdf |
| spelling | umjimathkievua-article-33792020-03-18T19:52:51Z Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series Тополого-метричні властивості множин дійсних чисел з умовами на їх розклади в ряди Остроградського Baranovskyi, O. M. Pratsiovytyi, M. V. Torbin, H. M. Барановський, О. М. Працьовитий, М. В. Торбін, Г. М. We study topological and metric properties of the set $$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$ with certain conditions on the sequence of sets $\{V_n\}$. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability theory. Исследуются тополого-метрические свойства множества $$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$ с определенными условиями на последовательность множеств $\{V_n\}$. В частности, установлены условия, при которых мера Лебега этого множества является: а) нулевой, б) положительной. Выполнено сравнение с соответствующими результатами для цепных дробей. Обсуждаются возможные применения полученных результатов в теории вероятностей. Institute of Mathematics, NAS of Ukraine 2007-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3379 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 9 (2007); 1155–1168 Український математичний журнал; Том 59 № 9 (2007); 1155–1168 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3379/3501 https://umj.imath.kiev.ua/index.php/umj/article/view/3379/3502 Copyright (c) 2007 Baranovskyi O. M.; Pratsiovytyi M. V.; Torbin H. M. |
| spellingShingle | Baranovskyi, O. M. Pratsiovytyi, M. V. Torbin, H. M. Барановський, О. М. Працьовитий, М. В. Торбін, Г. М. Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series |
| title | Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series |
| title_alt | Тополого-метричні властивості множин дійсних чисел з умовами на їх розклади в ряди Остроградського |
| title_full | Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series |
| title_fullStr | Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series |
| title_full_unstemmed | Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series |
| title_short | Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series |
| title_sort | topological and metric properties of sets of real numbers with conditions on their expansions in ostrogradskii series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3379 |
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