SIMULTANEOUS DIOPHANTINE APPROXIMATIONS AND A

!"
# $%# &'() +
* ,)
-/.%0,013254687:9,;0,7:681
<>=@?BA@CEDGFGAIHKJKL:MON,MONQPRMORTSVUXWTY:Z:[\SVWUX]_^\`ba@WNXMcSVWUX]_dc^e[\SVZ:UX^8Wf5^edc^eg^eRih
SVPUj]_dc^ek^edml5n!^8NXSVPUjSpoKMcSVLPq:UXMO^\frWsk^eUXkbMc^eoWfrNXWg^t[edcPNXNXMO[\PdGUj^\NjZidOSjN
MORvuwMcWa:L:PRTSVMcR:^_Paia:UjWs`TMcgPSVMcWRiNyxGNXZ:[zL{PN|uMcUjMc[zL:dc^eSy} N~SjL:^eWUj^\gl€n^
SVL:^\R Y:McNj[\Z:NXN|P[e^\UjSjPMcRvL:Mc‚L:^\U~Y:Mcgƒ^\R:NXMOWR:PdIPR:PdcW‚Z:^€WfpMcSBxrR:Pgƒ^\dO]
Pa:aiUjWs`TMcgƒPSVMcWR!Wf„a@WMcRTSVN8McR!P . Z:[edcMOYi^ePR!NjaiP[e^qT]a@WMcRTSVN8Wf%PZ:R:Mch
gWbYiZ:dOPU8dOPSVSjMO[\^bl|…mR!SjL:MON|Y:MOUX^e[\SVMcWRrx@o
^‚Mck^ƒPRWsk^\UjkTMc^eo3Wf%SVL:^f†Ph
gWZ:N‡[\WRyˆV^\[eSVZiUj^pWfЉMcR:‹WsoKNX‹TMbPR:YNjWg^%WfiSVLi^Kk^eUX]8^e`T[\MOSjMOR:‚tUX^e[\^eRTS
Y:^ek^edcWa:g^\RbSjNKWfKŒKlb‰[e‰Z:dOdc^eRWRSjL:McNQNjZiqTˆV^e[\SBl
Ž b‘b’@“”“K’‘b“•’G–B–\—”‘’G“Q˜B–B™š“œ›!m˜BžGŸ ’G¡š’G“œ¢%–B—•–B£

¤,¥§¦„¨:¥§© ªI«€¬®­E¯†°e±b²,­E³%²‡´/µ·¶s¸B¹»º­b¹¸¼E¸B²‡ºi¸!°m²½¹b¸s­E³ ºs¾€°e¿m¸À¿®Áb±b¹·¸BÂ/¸B¹»Ã½¯Åб
Æ3ÇÉÈÊ½Ë ¾!¯Å̄¸B¹¸v¸BÍÀ°mºi¯†ºvÎÏ ÊÑÐ ºs´pÒBÌ3¯Å̄­E¯
ÆÓÇÕÔ ÖÇ×È
Ø
¬¸Bµ·­b¹ÙÛÚG¿~Ü Ý› ™š–’@ÞG˜B›Àߚ’›®à®v’G“áஒy¢%–ƒà˜B—œ›À{˜\’›À—•žG¡š’G“¡§™šŸâ„b˜\–“”—•ã@ ÎÏ
—”¡“•žyà–V›®›Tb˜\Ÿ–\äK—”£•b£_ÞG‘å,æ Öiè8é Ú £
ê ߚ|–V›À’›ÀbŸ ¡À›žGi›Tߚp›ÀߚžGÔI˜Bç ŸŸ’G¡š–‡›Àߚ’›,¢šžG™‘’G¡’@ëšëš˜Bžyì%—•Ÿ’›À|’G¡T¢˜Bb’@“
¡§™šŸâ„b˜’G˜\⚗œ›À˜B’@˜B—•“á¢·à“”“wâÀ¢í˜B’›T—”ž@¡š’G“w¡§™šŸ!â%˜B–\£ïsஏeðšb˜›Àß5—”––ñ›À’›TbŸ b¡T›
—”–¡šž›ò%‘e›À—œð‡ä—•£”£—œ›"å5žr–!¡šž››Tb“•“~™š–!’@¡À¢§›Tߚ—”¡5ޒG℞G™Š››Àß5ë5˜BžG넏b˜ñ›À—•b–
žGp–\™š‘sߘB’›T—”žG¡5’G“”–ƒ¡/™5Ÿâ„b˜B–\£_ó„ž@˜ƒ—•¡š–V›T’G¡š‘bäpÞG—œð‡¡·’@¡—”˜\˜B’›T—”ž@¡š’G“„¡/™5Ÿâ„b˜_ôä
–B™šë5ë%žG–\›Àߚ’›_ஏà’G¡T››Àž ’Gëšë5˜Bžyì%—”Ÿ ’›T!—œ›âT¢õ˜B’›T—”žG¡5’G“p¡§™šŸâ„b˜\– ÎÏ à—á›Tß
ö Ö ö@÷ùø
mžG˜v–Bž@Ÿë%ž@–B—œ›À—œð‡·—”¡T›ÀÞGb˜ ø £úyà&‘b“•žG–B“á¢û‘’G¡üஏ’@ëšëš˜Bžyì%—•Ÿ’›Àô—•¡
›Àߚ—•–8‘b’@–Bb䄒G¡šå ߚžyàúŸ’@¡À¢–B™š‘sßï’Gëšëš˜\žsì%—•Ÿ ’›À—•žG¡š–~’G˜\w›Àߚ˜Bbý ê ߚ—”–~þ§™š–V›À—•žG¡
—”–ƒ›Tߚ žG˜B—•ÞG—•¡š’›T—”¡šÞ넞G—•¡À›ž@K›Tߚ›ÀߚžG˜V¢ÿžG
 —”žGë5ߚ’G¡T›À—•¡š ’Gëšëš˜\žyì
—•Ÿ ’›À—•žG¡š–\£
›à®’G
– š˜B–ñ›‘bžG¡5–B—•嚏b˜\båõâÀ
¢ —”˜\—”‘sß5“”Ý›—”¡ Ú ä„àß5ž·ë5˜Bžyð‡å›TߚvmžG“•“”žyà—•¡šÞ
‘b“”âš˜B’›Tbå ›ÀߚžG˜BŸ£
¤,¥§¦„¨:¥§© æ —•˜B—•‘sߚ“•e›Tä Ú è « w¸À¯ ô Ê½Ë ¶s¸­b²°m¹»¹b­E¯†°e±b²,­E³G²‡´rµ·¶s¸B¹»¾>­b²,¼
³†¸À¯ ø ¿
„
„
Ì
B
¸
¹

¸
B
¸
À
Í
m
°
i
º
%
¯
¹
À
¸
†
³
E
­
Å
¯
°mÂ/¸À³ à „¹°mµ·¸°m²,¯V¸ /¸B¹»º Ö Ä8°e¯†ÌÚ ÷ Ö ÷ø
Ê ÔIç
ºs´pÒBÌ3¯Å̄­E¯
ö ô Ö ö Ç Ú Ø
æÚè
Ô ø
!#"$"&%('*)$+-,/.$0&"1,2+3%
4 ±b¹¸s±bÂ/¸B¹»¾_¯†Ì„¸B¹¸v­b¹b¸®°m²5,²‡°e¯V¸À³ à µ·­b²‡Ãï¹­E¯†°e±b²,­E³r²‡´/µ·¶s¸B¹º ÎÏ ºs´
ÒBÌü¯†Ì„­E¯
æ
è
Ú
ô Ô Ö Ç 7Ö 6 Ø
¬¸Bµ·­b¹Ù8
I¿ ê ߚ>뚘Bž/žGšž@Kæ Ú è Ÿë5“”žy¢%–
›Àߚ|m’GŸ žG™š–9—”˜\—”‘sß5“”Ý›€â„žsìï뚘\—”¡5‘b—•ëš“”ä
¡š’GŸ b“œ¢õ›Àߚï—”嚏’ï›Àß5’›—•¢‡ž@™ÿߚ’yð‡;:Ñ℞yì%b–{’G¡5å=< >?:ўGâ#@e‘e›À–›Àžë5™Š›
—”¡T›Tž›Àß5bŸ äI›Tߚb¡’›!“”’G–V›{žG¡šâ%žyìõŸ™5–V› ‘žG¡T›À’G—•¡’›“•b’G–ñ››zஞžGâ#@e‘e›T–BAw—•¡
m’G‘e›T䄛Tߚ—”–—•–>›Tߚ›Àß5bžG˜\bŸ ›Tߚ’››Tߚ{뚘\—”¡5‘b—•ëš“”vÞ@ž› —œ›À–¡š’@Ÿvm˜\žGŸ £ C®ž›T—”‘
›Àߚ’›"æ è mžG“•“•žsà–_m˜\žGŸ&æ Ú Bè D 嚗œð%—”嚏℞›ÀߖB—•嚏–žGƒæ Ú è âÀ¢ Ö är›Tߚb¡ àߚ’yð‡
Ú
Ú
æ E è
ô Ô Ö Ç ø Ö ÷ FÖ 6
ç
Ö
–B—•¡š‘b ÷ø £ C®žyà –B™5ëšë%ž@–B€›Àߚ’›_›Tߚb˜\!’G˜\žG¡š“œG
¢ š¡š—œ›À“á¢õŸ ’G¡T¢õ˜B’›À—•žG¡š’G“•–
›Àߚ’›–B’›T—”–\c¢Ûæ è äp‘b’G“•“5›Àß5bŸ
Ü e›
ÔIÖ H TØ ØTØ ÔKÖ J Ø
ç ç
H
J
L é Ÿ —•¡ ô *Ô Ö O
ç
HNMOPMJ
O
›Àߚ¡ L >RQGäp–\—”¡š‘ôÿ—•–_—”˜\˜B’›T—”žG¡5’G“”£ Ü Ý› ø Ê â%–B™5‘sߛÀß5’›
Ú
Ç LØ
S ¢ æTE è är›Tߚb˜\Ÿ™š–ñ›{Ýì
—•–V› ÎÏ à—œ›Àß Ú ÷ Ö ÷ùø
ø
–\™š‘sߛTߚ’›
Ú
ô Ô Ö Ç ø ÖÇ L
ç
Ï Ê U ÎXÏ W ØbØTØ ÎÏ Y ä>àߚ—•‘sß —”–’õ‘bžG¡T›À˜\’G嚗•‘b—œ›ÀžG¡5£ ê ß/™5–›Àß5b˜B·Ÿ™5–V›
ߚb¡5‘b ÎV
Y
W
â%"—”¡Z5¡š—á›Tb“œ¢ Ÿ ’Gç ¡T¢ ç–\™š‘sßɘB’›T—”ž@¡š’G“•–B£[C®ž›À—•‘b!›Tߚ’›{›Tߚ—”–’G˜BÞG™5Ÿ¡T›·—”–ðšb˜ñ¢
–B—•Ÿ —”“•’G˜t›À]
ž \~™š‘b“•—•åZ^”–ëš˜Bž/žGQžG‡›Àߚ—•¡Zš¡š—œ›À™5嚏!ž@ëš˜\—”Ÿ –B£
 _£wót£tސž›Tß׿ Ú `aa è ätmž@˜>àߚ—•‘s߽ߚv˜\b‘b—œð‡b归
ê ߚ m’GŸ žG™š–ƒ›Tߚbž@˜BbŸ žG
ót—•b“•åš–®Ÿ bå5’G“w—•¡ Ú `a ä–ñ›À’›Tb–>›Tߚ’››Àߚ!eì%ë%ž@¡šb¡T›—•¡Éæ è —”–â%–V› 넞G–\–B—•âš“”ä
—”£•b£ —” cÏ H b —•¡õ›Àß5™šë5ë%˜ï℞G™š¡šå —”–˜Bëš“•’G‘bå âT¢ cÏ H d mžG8
˜ ef> ä>›Àߚ¡õ›Tߚ
¡§™šŸâ„b˜ž@p–BžG“•™Š›T—”žG¡5–ƒ‘b’@¡â%g
 š¡š—œ›À A
—•¡m’G‘e›Tä%—œ›—•
– 5¡š—á›T®mžG˜>eðšb˜V¢’G“•ÞGbâ5˜B’G—•‘
ô£
ê ߚ˜B×—”–ü’ “”’G˜\ÞGðr’G˜B—•e›z¢ ž@ïߚ—•ÞGߚ˜ü嚗”Ÿ ¡š–B—•žG¡š’G“ ÞG¡šb˜\’G“•—ih’›À—•žG¡š–õžG
—”˜\—”‘sß5“”Ý›j^•–›Àß5bžG˜\bŸ –B£lk  à—”“•“‘bžG¡š–\—”å5b˜žG¡5žG|›ÀߚŸ£lkۏ–ñ›À’G˜ñ›"à—œ›Àß
–BžGŸ !⚒G–B—•‘{¡šž›À’›T—”žG¡5£ Ü Ý›]mon ä’G¡5åӓ”Ý›qp ØbØTØ p-r Ê×Ë r ℏ“”—•¡šb’@˜B“œ¢
Hyç ç
s t "$+1uwv;,/+ tNx ,zy{u|"~}!&y€{)1*!
‚
r
—”¡5嚏b넏b¡šå5b¡T›ðšb‘e›TžG˜B–\£„ƒ†…‡|ˆ‰ˆ‰ŠT‹ ¥fŒ —”¡ Ë –\뚒G¡š¡šåíâT¢3›Àߚ–Bð‡b‘Ý›ÀžG˜\–—•–
›Àߚm˜\b 9 Ÿ ž/嚙š“•
r
é
é
D
Œ
–\뚒G¡ZŽj ØTØbØ 9rg‘
“ p “ ØTØTØ “ r Ê Ø
yH ç ç
O O yH ç ç
OP’{H
Ü e›
” é iæ p ØTØbØ p-r è
H
â%€›Àß5!‘žG˜B˜\b–B넞G¡šå5—”¡šÞ(mo•m ¡šžG¡š–\”— ¡5ÞG™š“•’G˜–—‡Z˜‰ŠT˜ © ‡Zˆ ¨ ŠT™ÿžG Œ £zš€“•b’@˜B“œ¢
⚒G–\—”–—•–¡šž›v™š¡š—•þ/™5bä~ߚb¡5‘bv¡š—á›Tߚb˜—•–âš’G–\—”–Ÿ ’›T˜B—œì
£î®žyஏeðšb˜Bä~—•3” ’G¡šå
› ’G˜Bƒ›zஞ"嚗•ò„b˜B¡T›{⚒@–B—•–Ÿ ’›T˜B—•‘b–B䚛Àߚ¡›Tߚb˜\Ýì%—”–ñ›À–gœ ʝŸž r æ è 䧛Tߚ
ÞG˜Bž@™šë{žG‡’G“•“Zm• C —•¡T›ÀÞG˜B’G“rŸ ’›À˜\—”‘b–
à—œ›Àßv嚏ݛÀb˜\Ÿ —”¡š’@¡À›þ/™5’G“i›Tž¡ Ú ä‡–B™š‘sß
›Àߚ’›Ÿ› é ”¢
œ äQ’G¡šå–Bž
ö 嚏e›§æ› è ö é ö 嚏ݛ/棔¢œ è ö é ö 嚏ݛ/棔 è öKö 嚏ݛ§æœ è ö é ö 嚏ݛ§æ£” è ö Ø
ê ߚ—•–®‘bž@ŸŸ žG¡ð5’@“”™š—•–®‘b’@“”“•b奤 ¥ ˆ ¥§¨:© ŠT¦9‡|¦§ˆ žG Œ äQ’G¡šåå5b¡šž›TbåÓå5e›ræ Œ è £
ƒ “”’›\›À—•‘b Œ —•–|‘b’Gr “•“”å©¨—¦9r Š ©Ñ¦ ¤9¨—…T‡ ¨ —•
å5e›ræ Œ è8é Ú £ ê ߚ–\—”Ÿ 뚓•b–ñ›eì%’GŸ 뚓•
žG,’“•’›\›À—•‘b—”¡ Ë —”– AK—œ›—•–‘“”’G˜B“œ¢ü™š¡š—•Ÿ žrå5™š“”’@˜B£
ª ™šë5ë%žG–\|ஏ®ßš’y𚏒Ÿiì
åë%ž@—”¡T›¬« Ê½Ë r £€î®žyà‘“”ž@–Bb“œ¢‘b’G¡à’Gëšëš˜\žsì%— 
Ÿ ’›À>—á›âÀ¢"넞G—”¡T›T–tžG r ý­®
C ž›À—•‘b›Tߚr ’›€mž@˜„›Àß5ƒëš™š˜\ë%ž@–Bb–8ž@‡’Gëšëš˜\žsì%—•Ÿ ’›À—•žG¡
à¡šb咟 Ý›À˜B—•‘b£óIžG˜’G‘sß®p
äK嚏š¡š
ʽË
¯ °æ p è8é ö Æ ØTØTØÝÆ r ö Ø
H
C®ž›T—”‘ ›Tߚ’››Àߚ—•–‘b’G¡ ℏv›ÀߚžG™5ÞGßT›ÓžG®’G–ïž@¡šë„žG–\–B—•âš“”ÞGb¡5b˜B’@“”—°hb’›T—”ž@¡ r Gž 
˜BÞG™š“•’G˜>’@⚖BžG“•™Š›T>ðr’G“”™5 ö
ö žG¡ Ë ät—•¡ëš’@˜V›À—•‘b™5“”’G˜>—á›—”¡5嚙š‘b–’{Ÿ e›T˜B—•‘žG Ë £
ê ߚb¡·—œ›—•–_¡šž›åš—°± ‘b™š“œ››Àž –B_›Tߚ’›
–B™šë —•¡šŽ  ¯ °æ « p è~é K· r Ø
²³X´wµ ¶³ µ
z¡ž›Àߚ˜tàžG˜\åšr –Bä5›Tߚb˜\eì%—”–ñ›{넞G—•¡T›À–_—”¡ Ë r ›Àߚ’›‘b’G¡š¡5ž›!â%’Gë5뚘Bžyì%—”Ÿ ’›Tbå
âT¢"넞G—•¡T›À–,žG ℏe›\›À˜I›Tߚ’G¡{™šë_›Àž 6 H àœ— ›Àß{˜\b–\ë%‘e›w›Tž|›TߚƒŸ e›T˜B—•‘ƒ—•¡šåš™š‘bå
µ
âT¢ ¯ £ z¡5嚏båšäQ—”
« é
›Àߚ¡—•¡·m’G‘e›
6H
6H
£
££
ʽË
r
ç
—”¡šŽ  ¯ æi« p è8é ¯ æi« ¢¸
è~é ¯ æi« è8é K· r Ø
¶³ µ
š€’G¡à 嚞ℏe›\›Àb˜—•Kàv˜\b뚓•’G‘b r à—œ›Àßüë%˜Bߚ’@뚖’G¡šž›Tߚb˜™š¡5—”Ÿ ž/嚙š“•’G˜
“”’›B›À—•‘býº¹Ó—•¡šã@žyà–\ã/—
‘bžG¡@e‘e›T™š˜Båv›Tߚ’›®’›“•b’@–V›ƒà®>à—•“”“%¡šž›®åšž{’G¡T¢àž@˜V›Àß5£
»
!#"$"&%('*)$+-,/.$0&"1,2+3%
Xš¡5’ ÞG˜\žG™šëžGåš—”’@ÞGžG¡š’G“
Ÿ’›À˜B—•‘b–
ØTØTØ Q
£
D >½Q
é Ú
£ £ £ £ ££
“
“ ØTØTØX“ r
Ø
O
ç
H
Q ØTØTØ “ r
# ¦ ¦{¾ ¥ ‹*ˆ‰¨ ¨:¥ÿªI«¿w±b¹­b²‡Ãï´/²‡°mµ·±À¼b´
³†­b¹³†­E¯j¯†°eÒs¸ÀŒ¢Á Ë r t
¾ и®Ì„­bÂ/¸jÂ
æ è
–B™šë —•¡š ¯ iæ « p è ÷ K· r Ø
¼ é
“
££ H
²³X´wµ ¶³7Ã
ÄgÅ ´p­E³ °e¯†Ãï̄±E³†¼bº®°cÁ­b²,¼±b²,³ Ãï°cÁ¯Å̄¸B¹¸v¸BÍÀ°mºi¯†ºqÆ Ê ¼ ºs´pÒBÌõ¯†Ì„­E¯
Œ é Æ r é –B뚒@¡ Ž  “
ØTØbØ “ r rg‘
HÈÇZyH ç ç Ç r ç
Ä8̄¸B¹¸
ØTØTØ r ­b¹¸¯†Ì„¸8ºi¯j­b²,é ¼E­b¹¼¶s­bºs°mº>Â/¸sÒÀ¯V±b¹»º~°m² Ë ¾t­b²,¼ “ ØTØTØ “ r Ê
Z
Ç
H
\
ç
HBç ç
ÚG¿
Ë ­b¹b¸-
±bºs°e¯†°mÂ/çɸÇ Ä8°e¯†Ì “ ØbØTØX“ r
¹ —•¡šã@žyà–\ã/—°•^ –‘bžG¡Êe@ b‘eH ›T™š˜Bßš’@–ℏb¡ ë5˜Bžyð‡å mžG˜]m é âT¢Ë¹Ó—•¡šã@žyà–B㧗
Ó
ߚ—•Ÿ–\b“•mäpmžG˜m é {
E âT¢ÓސbŸ ’G㗔¡ ÚF`
䄝mžG˜¬m é âÀ¢=ƒ¢%–BžG¡—•¡ Ú` ä%mž@˜
é
a âT¢ ª 㧙šâ„b¡šãŠžï—•¡ Ú`ÌÍ ä
’G¡šåmžG˜¬m é Í âÀ¢ºÓ
m
¹ ‘X
¹ ™š“”“•b¡—”¡ QQ a £9Û
k 
à—”“•“~⚘B—•XiÎ ¢×嚏b–\‘b˜B—•ℏ(Ó
¹ ‘Ó
¹ ™š“•“”¡Z”^ –eì%‘b—œ›À—•¡šÞӒGë5뚘BžG’@‘sß·›Àžv›Àß5—”–m’G–B‘—”¡5’›À—•¡šÞ
뚘Bž@⚓”Ÿ £óIžG˜w›Tߚ—•–8ஏ¡šbå–\žGŸ !Ÿ žG˜\¡šž›À’›À—•žG¡š£
óIžG˜€b’@‘sß ðšb‘e›TžG˜Ïp ÊÑË r ärà˜\—á›T
Ðp Ð é
r
6 ÒH Ñ
Æ
O
6
OP’{H
mžG˜€›Àߚï™š–B™š’@“Ï\~™š‘“”—•嚏b’@¡ÿ¡šžG˜BŸ&žG~p_£õóIžG˜’Ó“•’›B›T—”‘ ŒÓÁ Ë r >
ä 嚏Xš¡5—á›T–
‹ |¦ ÔI¥§¨ Ц—Õ ¨ ‡Z¤—ŠT¨9˜
Ö æ Œ è8é –B™šë —•¡š Ð « p Ð
æa è
Ø
²Ê³X´ µ ¶ ³7Ã
C®ž›T—”‘®›Àߚ’››Àߚ—•–—”–›Àß5–\’GŸ  e ì%뚘B–B–\—”žG¡½’G–—•¡ æ è ä8⚙i›à—á›Tß½˜\b–B넏b‘Ý›!›Tž
Ð×Ð  ¡šžG˜\Ÿ —•¡š–V›Tb’GåûžG ¯ £ØS¢ À› ß5õ—•¡šþ/™5’G“”—œ›z¢Õℏe›zஏbb¡û’G˜B—œ›Àß5ŸÝ›À—•‘ü’G¡šå
ÞGbž@ŸÝ›À˜\—”‘Ÿ b’G¡5–Bäšàߚ’yðš
æÍ è
¯ æ°p è µ W ÷ Ð p Ð
Ù m ç
mžG˜€’G“•“{p ÊÑË r £
ƒ “”’›B›À—•‘b Œ —•–~‘b’G“•“•båÚ ¥ …T…Û ¨:¦ ¨9¦—¤ ¥ ¤½—•§›Tߚb˜\Ýì%—”–ñ›m “•—”¡5b’G˜\“ᢷ—•¡šåšë%¡ 
嚏b¡T›~ð‡‘e›TžG˜B–|—”¡ Œ ߚ’yð%—•¡šÞ_›Tߚ_–B’GŸ \~™š‘b“•—•嚏b’G¡¡šžG˜\Ÿ bþ§™š’@“@›Àž—”¡5 ¶³Èà Рp Ð
—”£•b£’G“”“~–\ߚžG˜V›Tb–ñ›“•—”¡š’G˜B“œ¢×—•¡šåšb넏b¡5嚏b¡T›{ðšb‘e›TžG˜B–žG Œ ߚ’yð‡›Àߚ–B’GŸ 8\~™  ç
‘b“•—”å5b’G¡É¡šžG˜\Ÿ £ ê ß5mžG“”“•žyà—”¡5Þ3â%žG™5¡šåÿžG¡›Àß5‘bžyð‡˜B—•¡šÞü˜B’G嚗•™š–vߚ’@–vâ%b¡
뚘Bžyðšbåmž@˜wà“”“  ˜\žG™š¡šåšå“•’›B›T—”‘b–tàߚº
¡ m ÷ Í£
st "$+1uwv;,/+ tNx ,zy{u|"~}!&y€{)1*!
Ü
r
¤,¥§¦„¨:¥§© Ý æ£k۞/žr嚖\ä Ú`Ì
è « ¸À¯ m ÷ Í@¾{­b²,¼ü³†¸À¯-ŒÞÁ Ë
¶s¸­Ä¸À³m³ ß
¹±b´r²,¼E¸s¼!´/²‡°mµ·±À¼b´
³†­b¹³†­E¯j¯†°eÒs¸À¿8„ ̄¸B²
Ö æŒ è ÷ Ù m
Ø
ÄgÅ ´p­E³ °e¯†Ãï̄±E³†¼bº®°cÁ­b²,¼±b²,³ Ãï°cÁ Œ éRà r Áb±b¹ºi±bµ·¸ à Êáâ ræ Ë è ¿
¹Ó‘Ó
¹ ™š“•“”¡šäKžG¡!›Tߚ®ž›Tߚb˜ƒßš’G¡šåšä%뚘Bžyðšbå›Àß5’›’G¡T¢“”’›B›À—•‘b‘b’G¡â„®’@ëšëš˜Bž 
뚘B—•’›Tb“œ¢äÝ㠘B–B‘b’@“”åZ
å ›Tž’và“”“  ˜\žG™š¡šåšåížG¡š£=Ó
¹ ž@˜B–B넏b‘b—°š ‘’G“”“œ¢Eä,àïߚ’yð‡
›ÀߚmžG“•“•žsà—•¡šÞG£
¤,¥§¦„¨:¥§© æ æT¹Ó‘X¹Ó™5“”“•b¡šä QQ a è «¬¿w±b¹­b²‡Ãõ³†­E¯V¯Å°eÒs¸çŒèÁ Ë r ¾¯Å̄¸B¹¸¸BÍÀ°mºi¯†º
Æ Ê ¼ ºs´
ÒBÌõ¯Å̄­E¯|¯Å̄¸v³†­E¯V¯†°eÒs¸zÆgŒ °mºÄ¸À³m³ zß ¹b±b´/²,¼E¸s¼E¿
C žyà䇑žGŸâš—•¡š—•¡šÞ ê ߚžG˜BŸ à—œ›Àß·æ Í è äpæ a è 䚒G¡šå ê ß5bžG˜\bŸÞ|
®
E ¢
—•b“•åš–~€
š ž@¡ 
@eb‘Ý›À™š˜\ Ú mžG˜é ÷ Í £Ñb¡5‘b®Ó
¹ ‘Ó
¹ ™š“•“”¡Z•^ –m™5¡šåš’GŸ b¡T›T’G“ƒ‘žG¡T›À˜B—•âš™Š›T—”ž@¡3›Tž
¹ —•¡šã@žyà–B㧗°”^ –_‘žG¡e@ b‘Ý›À™š˜\>ஒG–›Àß5™š¡šåšr ˜B–ñ›À’G¡šå5—”¡šÞvž@mä%—•38 Ÿ ’s¢–B’y¢GäÝã 嚗•–V›T˜B— 
Ó
⚙Š›T—”žG¡|åž@Iஏb“•“  ˜BžG™5¡šåšb哔’›\›À—•‘b–—•¡ Ë £
ñ DÀAò ë í DÀA†ó F@?jô9õ ëÉö D§?<÷ ñùø î ó£ú ë CE?B†ó Aû&ô3
õ <Ïñ©øýüÉüÉþÉÿ|ô
ê¬ëÉì D§CTAí ëÉî A„ï#ð(
ï ë |
ë AbDÀAó£ï î ôKõ ëÉö D§?Éÿ Êü TüÉüÉþÊÿ
"!$#&%$%$'$($)&*$+$,.-+&*$)$/.-0#$1&/