!
!"#$%&'( !!)$* %#+
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!"#$ !"%&"#%'
"#$ α %# &#'( )&&'$)*+'( +,-%#&. /0# 1,+2$)*+ µα(t) )3 4#5+#4 '3 1*((*63.
/0# "#7#+4&# $0#*&#- 3$'$#3 $0'$ )1 α − A Q < 1 2Q2, (A, Q) = 1 89: $0#+ $0# 1&'2$)*+ A
Q )3 ' 2*+;#&7#+$ 1&'2$)*+ 1*& $0# 2*+$)+,#4 1&'2$)*+ #<='+3)*+ *1
α. /0# 2*+;#&3# 3$'$#-#+$ )3 +*$ $&,#. >$ -'? 0'==#+ $0'$ A
Q )3 ' 2*+;#&7#+$ $* α
%,$ 89: )3 +*$ ;'()4. @+# 30*,(4 2*+3)4#& $0# 3#A,#+2# *1 $0# 4#+*-)+'$*&3 *1 $0# 2*+;#&7#+$3 $* α 1*& 60)20 89: )3 $&,#. "#$ $0)3 3#A,#+2# %#
Q0 < Q1 < · · · < Qn < Qn+1 < · · · . /0#+ 1*& α 6∈ Q $0# 1,+2$)*+ µα(t) )3 4#5+#4 %? µα(t) = Qn+1− t Qn+1− Qn · ||Qnα|| + t − Qn Qn+1− Qn · ||Qn+1α||, Qn6t 6 Qn+1.
B&*- $0# *$0#& 0'+4C 1*& #;#&? ν *+# *1 $0# 2*+3#2,$);# 2*+;#&7#+$ 1&'2$)*+3 pν
qν, pν+1 qν+1 $* α 3'$)35#3 89:. D* #)$0#& (Qn, Qn+1) = (qν, qν+1) 1*& 3*-# ν C *& (Qn, Qn+1) = (qν−1, qν+1) 1*& 3*-# ν .
E2$,'((? $0# 1,+2$)*+ µα(t) 6'3 2*+3)4# %? F)+G*63G) HIJ. /0#&# #<)3$3 '+
'($#&+'$);# 7#*-#$&)2 4#5+)$)*+ *1 µα(t). D*-# &#('$#4 1'2$3 6#&# 4)32,33#4 )+ HKC LJ.
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
*+, -./01213
m
(α) = lim sup t→+∞ t · µα(t). 4/5 670528,9,8 20 :;<# "0 ,=>?2621 @79A.?/ @79 1+, B/?., 7@m
(α) 20 1,9A5 7@ 670120.,8 @9/61270 ,=>/05270 @79 α 4/5 >97B,8 20 :;<# C1 25 /5 @7??745# D.1m
n(α) = ( G(α∗ ν, α−1ν+2), 2@ (Qn, Qn+1) = (qν−1, qν+1) 421+ 57A, ν, F (α∗ ν+1, α−1ν+2), 2@ (Qn, Qn+1) = (qν, qν+1) 421+ 57A, ν, EFG 4+,9, G(x, y) = x + y + 1 4 , F (x, y) = (1 − xy)2 4(1 + xy)(1 − x)(1 − y) /08 αν, α∗ν 67A, @97A 670120.,8 @9/61270 ,=>/05270 17 α = [a0; a1, a2, . . . , at, . . . ] 20 5.6+ / 4/3H αν = [aν; aν+1, ...], αν∗ = [0; aν, aν−1, ..., a1]. *+,0m
(α) = lim sup n→+∞m
n(α), *+, 5>,691.A M= {m ∈ R : ∃α ∈ R \ Q 5.6+ 1+/1 m =m
(α)}. 4/5 51.82,8 20 :;<# C1 4/5 >97B,0 1+,9, 1+/1 M ⊂ 1 4, 1 2 /08 1+/1 1 4, 1 2 ∈ M# I74,B,9 07 @.91+,9 519.61.9, 7@ 1+, 5>,619.A M 25 J0740# C0 1+25 >/>,9 4, 670528,9 1+, 5>,619.A I= {m ∈ R : ∃α ∈ R \ Q 5.6+ 1+/1i
(α) = m}, 4+,9,i
(α) = lim inf n→∞m
n(α) E+74,B,9K 67A>/9,8 17m
(α)K 1+25 -./01213 +/5 07 6?,/9 L27>+/0120, 5,05,G# C1 25 6?,/9 1+/1 min I = 1 4, max I = 1 2. !"#$"%& !"#" "$%&'& ()&%'%*" ω0 &+,! '!-'1 4, ω0
⊂ I.
!"#$%& ! "#$%&'&( )*+,-%. )*+ ω0 ,./ 0" *0(.&!"1 )+*, (2" $+**) 0"%*34 5( &6
&!("+"6(&!7 (* 7"( *$(&,.% "6(&,.("6 )*+ (2" 8.%-" *) ω04
,5 6557 89:5 ;5<< =69;6 >58?<@82 &5AB<< @C5 75D6E@E96 9F B τ G85@ F ⊂ R2 %C5 85@ F :?8@ H5 9F @C5 F9>: F = S \ ∞ [ ν+1 ∆ν ! , ;C5>5 S ⊂ R E8 B 85I:56@J B67 ∆ν ⊂ S, ν = 1, 2, 3, ... E8 B6 9>75>57 85K?56A5 9F 7E8L9E6@ E6@5>MB<82 (9>59M5> F9> 5M5>N t EF S \ t−1 [ ν+1 ∆ν ! = r [ j=1 Mj E8 B ?6E96 9F 85I:56@8 Mj B67 ∆t ⊂ Mj∗ @C56 Mj∗ = N 1 ⊔ ∆t⊔ N 2 , B67 min(|N1 |, |N2 |) > τ |∆t|. $968E75> @C5 85@ F5 = {α = [0; b1, b2, b3, ...] : bν 65 ∀ ν}
A968E8@E6I 9F B<< E>>B@E96B< >5B< 6?:H5>8 F>9: @C5 ?6E@ E6@5>MB< (0, 1) ;E@C OB>@EB< K?9@E56@8 H9?6757 HN 52 )65 AB6 5B8E<N 855 @CB@
( A = min F5 = [0; 5, 1] = √ 45−5 10 = 0.1708 + , B = max F5 = [0; 1, 5] = √ 45−5 2 = 0.85410 + . P3Q "?@ S5 = [A, B] ⊂ [0, 1].
%C5 F9<<9;E6I <5::B A9:58 F>9: @C5 >58?<@8 9F @C5 OBO5>8 RST 9> RUT2 '!""# (& 92" 6"( F5 &6 . τ :6"( 3&(2 τ = τ5 = 1.788+4
15@ H(x, y) : S × S → R H5 B F?6A@E96 E6 @;9 MB>EBH<58 9F @C5 A<B88 G ∈ C1
(S × S)2 $968E75> @C5 85@
J = {z ∈ R : ∃ x, y ∈ S z = H(x, y)}. VN A96@E6?9?8E@N B>I?:56@ J E8 B 85I:56@2
'!""# )& ;-$$*6" (2.( (2" 1"+&8.(&8"6 ∂H/∂x, ∂H/∂y 1* !*( (.<" ="+* 8.%-"6 *! (2" 0*# S × S 4 ;-$$*6" (2.( F &6 τ :6"( .!1 S = [min F, max F]4 ;-$$*6" (2.(
τ > max x,y∈S max ∂H/∂x ∂H/∂y , ∂H/∂y ∂H/∂x . PWQ
!"# {z : ∃ x, y ∈ F $%&! '!(' z = H(x, y)} = J . $*++, - ./ , /0,.1203456,57 1*8*5,9.:,0.48 43 , 5*/;90 354+ <=># ?* 74 840 1.@* .0/ A5443 2*5* ,/ 02* A5443 349946/ 02* ,51;+*80 354+ <=> 6457BCDB6457# (46 6* ,5* ,C9* 04 E48E9;7* 02* A5443 43 F2*45*+# ?* E48/.7*5 A,.5/ 43 .80*1*5/ (R1, R2) 43 02* 345+ (R1, R2) = (R, R) 45 (R, R + 1) G=H 6.02 R > 6# I48/.7*5 , 3;8E0.48 HR1,R2(x, y) = F 1 R1+ x , 1 R2+ y .
J45 R1, R2 ;87*5 E48/.7*5,0.48 02* 3;8E0.48 HR1,R2(x, y) 7*E5*,/*/ C402 .8 x ,87
.8 y # J45 0 < x, y < 1 A;0 ϕ(x, y) = (1 − 3x + 3xy − x2 y)(1 − x). J45 ,8D y ∈ (0, 1) 02* 3;8E0.48 ϕ(x, y) 7*E5*,/*/ .8 x# J45 ,8D x ∈ (0, 1) 02* 3;8E0.48 ϕ(x, y) .8E5*,/*/ .8 y # (46 ∂F/∂y ∂F/∂x = ϕ(x, y) ϕ(y, x), ,87 ∂HR1,R2/∂y ∂HR1,R2/∂x = ϕ 1 R1+x, 1 R2+y ϕ 1 R2+y, 1 R1+x R1+ x R2+ y 2 . %,/D E,9E;9,0.48 /246/ 02,0 345 R1, R2 >6 48* 2,/ max x,y∈S5 max ∂HR1,R2/∂x ∂HR1,R2/∂y , ∂HR1,R2/∂y ∂HR1,R2/∂x = = ϕ 1 R1+B, 1 R2+A ϕ 1 R2+A, 1 R1+B R1+ B R2+ A 2 6 ϕ 1 R1+B, 1 R1+A ϕ 1 R1+A, 1 R1+B R1+ B R1+ A 2 6 6 ϕ 1 6+B, 1 6+A ϕ 1 6+A, 1 6+B 6 + B 6 + A 2 = 1.363+ < τ5. K*5* A ,87 B ,5* 7*L8*7 .8 G H ,87 .8 02* 9,/0 .8*M;,9.0.*/ 6* ;/* 02* C4;87/ 6 6 R1 6R2 62.E2 349946/ 354+ G=H#
,5 655 7897 :;< 9=> R1, R2 ?=@5< A;=6B@5<97B;= 9=@ :;< τ5C657 F5 785 A;=@B7B;= DEF B6 697B6G5@2 ,5 9HHI> 15JJ9 K 7; 655 7897 785 BJ9L5 ;: 785 657 F5× F5 ?=@5< 785 J9HHB=L HR1,R2(x, y) B6 M?67 785 65LJ5=7 JR1,R2 = [HR1,R2(B, B), HR1,R2(A, A)]. N?7 HR,R(B, B) < HR,R+1(A, A) 9=@ HR,R+1(B, B) < HR+1,R+1(A, A). %897 B6 O8> B: O5 H?7 ω0 = HR0,R0(A, A). OB78 R0 >6 O5 L57 [ R>R0 JR,R ∪ [ R>R0 JR,R+1 = (1/4, ω0]. %9P5 m ∈ (0, ω0]2 %85= 785<5 5QB676 R1, R2 6?A8 7897 m ∈ JR1,R2 9=@ 785<5 5QB67 β = [0; b1, b2, ..., bν, ...], γ = [0; c1, c2, ..., cν, ...], β, γ ∈ F5, 6?A8 7897 F 1 R1+ α , 1 R2+ β = m. .;O O5 79P5 α= [0; a1, R1, R2, b1 | {z } 1 , a2, a1, R1, R2, b1, b2 | {z } 2 , ..., aν, aν−1, ..., a2, a1, R1, R2, b1, b2, ..., bν−1, bν | {z } ν , ...]. !79=@9<@ 9<L?J5=7 68;O6 7897 :;< nν @5G=5@ :<;J pnν qnν = [0; a1, R1, R2, b1, a2, a1, R1, R2, b1, b2, ..., aν, aν−1, ..., a2, a1, R1] ;=5 896 lim ν→+∞F (α ∗ nν, α −1 nν+1) = m. 7 785 69J5 7BJ5 :;< F (·, ·) 9=@ G(·, ·) O5 89R5 inf n6=nν∀ν F (α∗n, αn+1) > ω0 9=@ inf n∈Z+ G(α∗n, α−1n+2) > ω0, :;< I9<L5 R02 !;
i
(α) = m 9=@ 5R5<>78B=L B6 H<;R5@2 ✷!" #$ %"&"'#()**#+ ,"&-')&.'/
*+, '# "-./0-1 234.56 -/.- 347 489:/6- ;<.= 6/-.6<>./7 ?36.<30 @85.</4.-1 A634-# "9/6# B3.=# '5>#1 CDE+ F+GGGH1 + I +JK#
*D, B# L300 FM6#H1 )4 .=/ -89 347 ?6578>. 5N >54.<48/7 N63>.<54-1 "4430/- 5N B3.=/93.<>-1 O!1 (5# O F+GOJH1 GPP I GG #
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˘c<8- /. 30# F%7-H1 .5 3??/36` ?6/?.<4. <- 3T3<03:0/ 3. 36a<TE+DKD#OPDDTD FDK+DH# *J, X#"# X<-36/T1 )4 .=/ -/. 5N 489:/6- 6/?6/-/4.3:0/ 3- >54.<48/7 N63>.<54- ;<.= :5847/7 ?36.<30 @85.</4.-1 Z8--<34 B3.=/93.<>30 '86T/Y-1 DKKK1 CCEC1 GG! I GGG R/?36.9/4. 5N B3.=/93.<>-1 "T/<65 ^4<T/6-<.Y1 "T/<65 !+K1 X56.8V30 bcdefghic D+#+D#DK++
