❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❙♦❜r❡ ♦ ♥ú♠❡r♦
π
†♣♦r
▼❛r❝❡❧♦ ❘♦❞r✐❣✉❡s ◆✉♥❡s ❉❛♥t❛s
s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
▼❛rç♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❙♦❜r❡ ♦ ♥ú♠❡r♦
π
♣♦r
▼❛r❝❡❧♦ ❘♦❞r✐❣✉❡s ◆✉♥❡s ❉❛♥t❛s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✳ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦ ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ▼✐❣✉❡❧ ❋✐❞❡♥❝✐♦ ▲♦❛②③❛ ▲♦③❛♥♦ ✲ ❯❋P❊
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r ♠❡ ♠♦str❛r ♦s ❝❛♠✐♥❤♦s ❞❛ ❧✉③ ❡ ❝♦♥❝❡❞❡r s❛❜❡❞♦r✐❛ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s q✉❡ ♣r❡❝✐s❡✐✳
❆ ❊r❝í❧✐♦ ❙♦❛r❡s ❉❛♥t❛s✱ ♠❡✉ ♣❛✐✱ ♠❡s♠♦ ♥ã♦ ❡st❛♥❞♦ ♠❛✐s ❡♥tr❡ ♥ós✱ t❡♠ ♣❛rt✐✲ ❝✐♣❛çã♦ ❛t✐✈❛ ❡♠ t♦❞❛ ♠✐♥❤❛ ✈✐❞❛ ❡ ❛ ▼❛r✐❛ ■r❛♥✐ ❘♦❞r✐❣✉❡s ◆✉♥❡s ❉❛♥t❛s✱ ♠✐♥❤❛ ♠ã❡✱ ♣♦r t♦❞♦ ❡♠♣❡♥❤♦✱ ♣♦r s❡r sí♠❜♦❧♦ ❞❡ t✉❞♦ q✉❡ ❤á ❞❡ ♠❛✐s s✉❜❧✐♠❡ ✱ ♣♦r t♦❞♦ ❛♠♦r✳ ▼ã❡✱ ❡ss❡ tr❛❜❛❧❤♦ é ♠❛✐s s❡✉ ❞♦ q✉❡ ♠❡✉✳
❆ ❚❛❧✐t❛ ❍❡❧❡♥ ❆r❛ú❥♦ ❞❡ ❆❧♠❡✐❞❛✱ ♠✐♥❤❛ ❡s♣♦s❛✱ ❡ ▼❛r❝❡❧♦ ❘♦❞r✐❣✉❡s ◆✉♥❡s ❉❛♥t❛s ❋✐❧❤♦✱ ♣♦r s✉❛s ❡①✐stê♥❝✐❛s✱ ♣♦✐s ❛♠❜♦s sã♦ ♠♦t✐✈♦s ❞❡ ♠❡✉ ♠á①✐♠♦ ❡♠♣❡♥❤♦ ♥❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❆ ✈✐❞❛ é ❜❡♠ ♠❡❧❤♦r ❝♦♠ ✈♦❝ês✳
❆ ▲❡♥✐♠❛r ◆✉♥❡s ❞❡ ❆♥❞r❛❞❡✱ ♣r✐♠♦✱ ♣❛❞r✐♥❤♦ ❡ í❞♦❧♦✳ ❙❡♠♣r❡ ❢♦✐ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ✈✐❞❛✳
❆ ▼❛r❝í❧✐♦ ❘♦❞r✐❣✉❡s ◆✉♥❡s ❉❛♥t❛s✱ ♠❡✉ ✐r♠ã♦✱ ♣♦r t♦❞❛ ♣❛r❝❡r✐❛ ❡ ❜♦♥s ♠♦✲ ♠❡♥t♦s✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧♦ ❝r❡s❝✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧ ♣r♦♣♦r✲ ❝✐♦♥❛❞♦ ❡ às ❝♦♦r❞❡♥❛çõ❡s✱ ❧♦❝❛❧ ❡ ❣❡r❛❧✱ ❞♦ P❘❖❋▼❆❚ ♣❡❧♦ ❡①❝❡❧❡♥t❡ ♣r♦❥❡t♦ q✉❡ ❞❡✉ ♥♦✈♦ r✉♠♦ à ♠✐♥❤❛ ✈✐❞❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❊❧♦♥ ▲❛❣❡s ▲✐♠❛✱ ♣♦r t♦❞❛ ❜♦♥❞❛❞❡ ❡♠ ♠❡ ♦✉✈✐r ♥❛ ❤♦r❛ ❡♠ q✉❡ ♣r❡❝✐s❡✐✳ ❙❡♠ s✉❛ ❛❥✉❞❛ ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✉✐❞♦✳ ❆ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r t♦❞❛ ❝♦♥tr✐❜✉✐çã♦ ✐♥t❡❧❡❝t✉❛❧ ❡ ♣❡❧❛s ❤♦r❛s ❞❡ ❡♠♣❡♥❤♦ ❡♠ ❜✉s❝❛ ❞❡ ♠❡ ♠♦str❛r ♦s ♠❡❧❤♦r❡s ❝❛♠✐♥❤♦s ❞❛ ♣❡sq✉✐s❛✳ ▼✐♥❤❛s ❡①♣❡❝t❛t✐✈❛s ❢♦r❛♠ ❛❧❝❛♥ç❛❞❛s ❡ s✉♣❡r❛❞❛s ♣♦r s❡✉ ❡①❝❡❧❡♥t❡ ❛♣♦✐♦✳
❆♦s ❛♠✐❣♦s P❡❞r♦ ❏ú♥✐♦r✱ ❋❡r♥❛♥❞♦ ❱✐❛♥❛✱ ❊❞s♦♥ ❋✐❧❤♦✱ ❋r❛♥❝✐s❝♦ ▲✐♠❛ ❡ ❈í❝❡r♦ ❉❡♠étr✐♦✱ ♣❡❧❛s ✈❡r❞❛❞❡✐r❛s ❛♠✐③❛❞❡s ❝✉❧t✐✈❛❞❛s ❣r❛ç❛s ❛♦ ❛❞✈❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ ♥♦ss❛s ✈✐❞❛s✳
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
P♦r ♠❛✐s ❞❡ ✷✺✵✵ ❛♥♦s✱ ♠✉✐t♦s ❞♦s ❣r❛♥❞❡s ♠❛t❡♠át✐❝♦s s❡ ✐♥t❡r❡ss❛r❛♠ ♥❛ ♥❛✲ t✉r❡③❛ ❡ ♥♦s ♠✐stér✐♦s ❞♦ ❢❛s❝✐♥❛♥t❡ ♥ú♠❡r♦π✱ ♠❡♥t❡s ❜r✐❧❤❛♥t❡s ❝♦♠♦ ❆rq✉✐♠❡❞❡s✱
❊✉❧❡r✱ ●❛✉ss✱ ❆❜❡❧✱ ❏❛❝♦❜✐✱ ❲❡✐❡rstr❛ss✱ ❡♥tr❡ ♦✉tr♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦ ♥ú♠❡r♦ π✳ ■♥✐❝✐❛♠♦s
♥♦ss♦ tr❛❜❛❧❤♦✱ ♣r♦✈❛♥❞♦ q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛r❜✐trár✐❛ ❡ s❡✉ ❞✐â♠❡tr♦ é ❝♦♥st❛♥t❡✳ P❛r❛ ✐st♦✱ ✉s❛♠♦s ❛ ❝♦♠♣❧❡t✉❞❡ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❚❛❧ ❝♦♥st❛♥t❡ é ♣r❡❝✐s❛♠❡♥t❡ ♦ ♥ú♠❡r♦ π✳ ❖ ❈❛♣ít✉❧♦ ✷ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦
❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ π✳ ❆♣r❡s❡♥t❛♠♦s três ♣r♦✈❛s✱ ❛ ❝❧áss✐❝❛✱ ❞❡✈✐❞❛ ❛ ▲❛♠❜❡rt✱ ❡
❞✉❛s ♣r♦✈❛s ♠❛✐s ♠♦❞❡r♥❛s ❞❡ ❈❛rt✇r✐❣❤t ❡ ■✈❛♥ ◆✐✈❡♥✳ ❆❧é♠ ❞❡ s❡r ✐rr❛❝✐♦♥❛❧✱ ♦ ♥ú♠❡r♦π é tr❛♥s❝❡♥❞❡♥t❡✱ ✐st♦ é✱ ♥ã♦ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s
r❛❝✐♦♥❛✐s q✉❡ t❡♥❤❛ π ❝♦♠♦ r❛✐③✳ ❚❛❧ ❢❛t♦ ❢♦✐ ❞❡♠♦♥str❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ▲✐♥❞❡✲
♠❛♥♥ ❡✱ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♦ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦ ♥ã♦ t❡♠ s♦❧✉çã♦✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s✱ s❡♠ ♣r♦✈❛✱ ✉♠ r❡s✉❧t❛❞♦ ♠❛✐s ❣❡r❛❧✱ ♦ ❝❡❧❡✲ ❜r❛❞♦ ❚❡♦r❡♠❛ ❞❡ ▲✐♥❞❡♠❛♥♥✲❲❡✐❡rtr❛ss q✉❡ t❡♠ ❝♦♠♦ ❝♦r♦❧ár✐♦✱ ❛ tr❛♥s❝❡♥❞ê♥❝✐❛ ❞❡ π✳ ❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❝❛♣ít✉❧♦ ✹✱ ❛ ❝r♦♥♦❧♦❣✐❛✱ ❝✉r✐♦s✐❞❛❞❡s✱ ❛♣r♦①✐♠❛çõ❡s ❡ sér✐❡s
s♦❜r❡ π sã♦ ❡st✉❞❛❞❛s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ π✱ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡✱ tr❛♥s❝❡♥❞ê♥❝✐❛✱ ❝r♦♥♦❧♦❣✐❛✳
❆❜str❛❝t
❋♦r ♠♦r❡ t❤❛♥ ✷✺✵✵ ②❡❛rs✱ ♠❛♥② ♦❢ t❤❡ ❣r❡❛t ♠❛t❤❡♠❛t✐❝✐❛♥s ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♥❛t✉r❡ ❛♥❞ t❤❡ ♠②st❡r✐❡s ♦❢ ❢❛s❝✐♥❛t✐♥❣ ♥✉♠❜❡r π ✱ ✇♦♥❞❡r❢✉❧ ♠✐♥❞s s✉❝❤ t❤❛t ❆r✲
❝❤✐♠❡❞❡s✱ ❊✉❧❡r✱ ●❛✉ss✱ ❆❜❡❧✱ ❏❛❝♦❜✐✱ ❲❡✐❡rstr❛ss✱ ❛♠♦♥❣ ♦t❤❡rs✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ st✉❞② s♦♠❡ ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♣r♦♣❡rt✐❡s t❤❛t ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♥✉♠❜❡r π✳ ❲❡
❜❡❣✐♥ ♦✉r ✇♦r❦✱ ♣r♦✈✐♥❣ t❤❛t t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ❧❡♥❣t❤ ♦❢ ❛♥ ❛r❜✐tr❛r② ❝✐r❝✉♠✲ ❢❡r❡♥❝❡ ❛♥❞ ✐ts ❞✐❛♠❡t❡r ✐s ❝♦♥st❛♥t✳ ❋♦r t❤✐s✱ ✇❡ ✉s❡ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs✳ ❚❤✐s ❝♦♥st❛♥t ✐s ♣r❡❝✐s❡❧② t❤❡ ♥✉♠❜❡r π✳ ❚❤❡ ❝❤❛♣t❡r ✷ ✐s ❞❡❞✐❝❛t❡❞ t♦
❤❡ st✉❞② ♦❢ t❤❡ ✐rr❛t✐♦♥❛❧✐t② ♦❢ π✳ ❲❡ ♣r❡s❡♥t t❤r❡❡ ♣r♦♦❢s✱ ❛ ❝❧❛ss✐❝❛❧ ♣r♦♦❢✱ ❞✉❡ t♦
▲❛♠❜❡rt✱ ❛♥❞ t✇♦ ♠♦❞❡r♥ ♣r♦♦❢s ❞✉❡ t♦ ❈❛rt✇r✐❣❤t ❛♥❞ ■✈❛♥ ◆✐✈❡♥✳ ■♥ ❛❞❞✐t✐♦♥ t♦ ❜❡ ✐rr❛t✐♦♥❛❧✱ t❤❡ ♥✉♠❜❡r π ✐s tr❛♥s❝❡♥❞❡♥t❛❧✱ t❤❛t ✐s✱ t❤❡r❡ ✐s ♥♦t ❛ ♥♦♥ ③❡r♦
♣♦❧②♥♦♠✐❛❧ ✐♥ ♦♥❡ ✈❛r✐❛❜❧❡ ✇✐t❤ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts t❤❛t ❤❛s π ❛s r♦♦t✳ ❚❤✐s ❢❛❝t
✇❛s ✐♥✐t✐❛❧❧② ♣r♦✈❡❞ ❜② ▲✐♥❞❡♠❛♥♥ ❛♥❞ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ♦❢ sq✉❛r✐♥❣ t❤❡ ❝✐r❝❧❡ ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ■♥ t❤❡ ❝❤❛♣t❡r ✸ ✇❡ ♣r❡s❡♥t ✱ ✇✐t❤♦✉t ♣r♦♦❢✱ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t✱ t❤❡ ❝❡❧❡❜r❛t❡❞ ▲✐♥❞❡♠❛♥♥✲❲❡✐❡rstr❛ss t❤❡♦r❡♠✱ ✇❤✐❝❤ ❤❛s ❛ ❝♦r♦❧❧❛r② ✱ t❤❡ tr❛♥s❝❡♥❞❡♥❝❡ ♦❢ π✳ ❋✐♥❛❧❧②✱ ✐♥ t❤❡ ❝❤❛♣t❡r ✹✱ ❝❤r♦♥♦❧♦❣②✱ ❝✉r✐♦s✐✲
t✐❡s✱ ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ s❡r✐❡s ♦♥ π ❛r❡ st✉❞✐❡❞✳ ❑❡② ✇♦r❞s ✿ π✱ ✐rr❛t✐♦♥❛❧✐t②✱
tr❛♥s❝❡♥❞❡♥❝❡✱ ❝❤r♦♥♦❧♦❣②✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ①
✶ ❉❡✜♥✐çã♦ ❡ ❡①✐stê♥❝✐❛ ❞♦ ♥ú♠❡r♦ π ✶
✶✳✶ ❆ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❆ ❡①✐stê♥❝✐❛ ❞❡π ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✸ ❈á❧❝✉❧♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ ✉s❛♥❞♦ ❧✐♠✐t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❚❡♦r❡♠❛ ❞❛ ❛❞✐çã♦ ♣❛r❛ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺ ❆♣r♦①✐♠❛♥❞♦ ❛ ár❡❛ ❞❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r ♣♦r ❞❡❢❡✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✻ ❆♣r♦①✐♠❛♥❞♦ ❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ♣♦r ❡①❝❡ss♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷ ❉❡♠♦♥str❛çõ❡s ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ π ✶✼
✷✳✶ ◆ú♠❡r♦s ♥❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❈♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸ Pr♦✈❛ ❞❡ ■✈❛♥ ◆✐✈❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹ Pr♦✈❛ ❞❡ ❈❛rt✇r✐❣❤t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✺ ❆ ♣r♦✈❛ ❞❡ ▲❛♠❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸ ❚r❛♥s❝❡♥❞ê♥❝✐❛ ❞❡ π ✸✻
✸✳✶ ◆ú♠❡r♦s ❛❧❣é❜r✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❆ ◗✉❛❞r❛t✉r❛ ❞♦ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸ ❚❡♦r❡♠❛ ❞❡ ▲✐♥❞❡♠❛♥♥ ✲ ❲❡✐❡rstr❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹ ❈r♦♥♦❧♦❣✐❛ ❡ ❝✉r✐♦s✐❞❛❞❡s s♦❜r❡ π ✹✵
✹✳✶ ❖ ♣❡rí♦❞♦ ❣❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✶✳✶ P♦❧í❣♦♥♦s ✐♥s❝r✐t♦s ❝♦♠ ♥ ❡ ✷♥ ❧❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷ ❖ P❡rí♦❞♦ ❈❧áss✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✶ ❙ér✐❡ ❞❡ ▲❡✐❜♥✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✷ ❙ér✐❡ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✸ ❙ér✐❡ ❞❡ ▼❛❝❤✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✷✳✹ ❖✉tr❛s ❋ór♠✉❧❛s ❡♥✈♦❧✈❡♥❞♦π ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✸ ❖ ♣❡rí♦❞♦ ♠♦❞❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✹ ❆♣r♦①✐♠❛çõ❡s ❞❡ π ✉s❛♥❞♦ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✹✳✶ ❆❣✉❧❤❛s ❞❡ ❇✉✛♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✹✳✷ ▼ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✺ ❈✉r✐♦s✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✺✳✶ ▼♥❡♠ô♥✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✺✳✷ ❉✐❢❡r❡♥ç❛ ❮♥✜♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✺✳✸ ◗✉❡♠ é ♠❛✐♦r✿ eπ ♦✉πe❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✺✳✹ ◗✉❡stõ❡s ❡♠ ❛❜❡rt♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
■♥tr♦❞✉çã♦
P♦r ♠❛✐s ❞❡ ✷✺✵✵ ❛♥♦s✱ ♦ ♥ú♠❡r♦ π t❡♠ ♦❝✉♣❛❞♦ ✉♠❛ ♣♦s✐çã♦ ❝❡♥tr❛❧ ♥❛ ❤✐stó✲
r✐❛ ❞❛ ♠❛t❡♠át✐❝❛✳ ▼✉✐t♦s ❞♦s ❣r❛♥❞❡s ♠❛t❡♠át✐❝♦s s❡ ✐♥t❡r❡ss❛r❛♠ ♥❛ ♥❛t✉r❡③❛ ❡ ♦s ♠✐stér✐♦s ❞♦ ❢❛s❝✐♥❛♥t❡ ♥ú♠❡r♦ π ✿ ❆rq✉✐♠❡❞❡s ❡ ♦s ♠❛t❡♠át✐❝♦s ❞❛ ❛♥t✐❣❛
●ré❝✐❛ ❝❛❧❝✉❧❛r❛♠ ✈ár✐❛s ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✳ ❖s ♠❛✲ t❡♠át✐❝♦s ❞♦ ❘❡♥❛s❝✐♠❡♥t♦ ❛❝❤❛r❛♠ ❛♣❧✐❝❛çõ❡s ♥❛ ❛rt❡ ❡ ❛rq✉✐t❡t✉r❛✳ ❊✉❧❡r ❡ s❡✉s ✐♥ú♠❡r♦s tr❛❜❛❧❤♦s✳ ●❛✉ss✱ ♣r✐♥❝❡♣s ♠❛t❤❡♠❛t✐❝♦r✉♠✱ ❡s♣❡❝✐❛❧✐st❛ ❡♠ ❝✐❝❧♦t♦♠✐❛✱ ✐st♦ é✱ ♥❛ ❞✐✈✐sã♦ ❞♦ ❝ír❝✉❧♦ ❡♠ ♣❛rt❡s ✐❣✉❛✐s ♦✉ ❛ ❝♦♥str✉çã♦ ❞❡ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ♦✉✱ ❛♥❛❧✐t✐❝❛♠❡♥t❡✱ ❛❝❤❛r ❛s r❛í③❡s ♥✲és✐♠❛s ❞❛ ✉♥✐❞❛❞❡ ❡ q✉❡ ♠✉✐t❛s ✈❡③❡s ❡st❡✈❡ à ❢r❡♥t❡ ❞❡ s❡✉ t❡♠♣♦✳ ❆❜❡❧ ❡ ❏❛❝♦❜✐✱ ❝r✐❛❞♦r❡s ❞♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ t❡♦r✐❛ ❞❛s ❢✉♥çõ❡s ❡❧í♣t✐❝❛s✳ ❍❡r♠✐t❡✱ ❝✉❥❛ ♣r♦✈❛ ❞❛ tr❛♥s❝❡♥❞ê♥❝✐❛ ❞♦ ♥ú♠❡r♦ e ✐♥s♣✐r♦✉ ❛ ♣r♦✈❛ ❞❛
tr❛♥s❝❡♥❞ê♥❝✐❛ ❞❡ π ♣♦r ▲✐♥❞❡♠❛♥♥ ❡ s✉❛ ♣♦st❡r✐♦r ❣❡♥❡r❛❧✐③❛çã♦ ♣♦r ❲❡✐❡rstr❛ss
❝♦♠ ♦ ❝❡❧❡❜r❛❞♦ ❚❡♦r❡♠❛ ❞❡ ▲✐♥❞❡♠❛♥♥✲❲❡✐❡rstr❛ss✳ ❊ ❘❛♠❛♥✉❥❛♥✱ ❝✉❥❛ ✐♥❡s♣❡✲ r❛❞❛ ❢ór♠✉❧❛ ❛✐♥❞❛ ❞❡s♣❡rt❛ ♦ ✐♥t❡r❡ss❡ ❞♦s ❡s♣❡❝✐❛❧✐st❛s ♥❛ ❛t✉❛❧✐❞❛❞❡✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦ ♥ú♠❡r♦π✳ P❛r❛
✐st♦✱ ❞✐✈✐❞✐♠♦s ❛ ❞✐ss❡rt❛çã♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ✉s❛♠♦s ❛ ❝♦♠♣❧❡t✉❞❡ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ♣❛r❛ ❞❡✜♥✐r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✳ ▼♦str❛♠♦s q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ s❡✉ ❞✐â♠❡tr♦ é s❡♠♣r❡ ❝♦♥st❛♥t❡✱ ✐st♦ é✱ ✐♥❞❡♣❡♥❞❡ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❚❛❧ ❝♦♥st❛♥t❡ é ♣r❡❝✐s❛♠❡♥t❡
π✳ ❚❛❧ ❢❛t♦ é ❜❡♠ ❝♦♥❤❡❝✐❞♦✱ ♠❛s✱ ♣♦✉❝❛s ✈❡③❡s ❞❡♠♦♥str❛❞♦ ♥♦s t❡①t♦s✳ ❋❛r❡♠♦s
✉♠❛ ♣r♦✈❛ ❡❧❡♠❡♥t❛r ❝♦♠ ❛❥✉❞❛ ❞❛ ❣❡♦♠❡tr✐❛✳ ❯s❛♥❞♦ ❢❛t♦s s♦❜r❡ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡✱ é ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳ ❯♠❛ ❞❛s ♣r✐♠❡✐r❛s ❝❛r❛❝t❡ríst✐❝❛s é q✉❡ π é ✐rr❛❝✐♦♥❛❧✳ ❆ ♣r✐♠❡✐r❛ ♣r♦✈❛ ❞❡st❡ ❢❛t♦ ❢♦✐ ♦❜r❛ ❞♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês
❏♦❤❛♥♥ ❍❡♥r✐❝❤ ▲❛♠❜❡rt ❡♠ ✶✼✼✼✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s três ♣r♦✈❛s ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ π q✉❡ ✐♥✲
❝❧✉❡♠ ❛ ❞❡ ▲❛♠❜❡rt✱ ▼❛r② ❈❛rt✇r✐❣❤t ❡ ✉♠❛ ♣r♦✈❛ ♠❛✐s ♠♦❞❡r♥❛ ❞♦ t❡♦r✐st❛ ■✈❛♥ ◆✐✈❡♥✳ ❯♠ ❞♦s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡ ❢♦✐ ❛ ❝❤❛♠❛❞❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦✱ ❝♦♥s✐st✐♥❞♦ ❡♠ ❝♦♥str✉✐r ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ❛ ♠❡s♠❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ❞❛❞♦✳ ❊♠ ✶✽✽✷✱ ❋❡r❞✐♥❛♥❞ ▲✐♥❞❡♠❛♥♥ ♣r♦✈♦✉ q✉❡ π é ✉♠ ♥ú♠❡r♦ tr❛♥s❝❡♥❞❡♥t❡✱
✐st♦ é✱ ♥ã♦ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❛❝✐♦♥❛✐s ♥ã♦ t♦❞♦s ♥✉❧♦s ❞♦s q✉❛✐s
π s❡❥❛ ✉♠❛ r❛✐③✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞✐st♦ ♦ ♣r♦❜❧❡♠❛ ❞❛ q✉❛❞r❛t✉r❛ é ✐♠♣♦ssí✈❡❧✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ❛s ♥♦çõ❡s ❞❡ ♥ú♠❡r♦s ❛❧❣é❜r✐❝♦s ❡ ❞❡♠♦♥str❛♠♦s q✉❡ sã♦ ❡♥✉♠❡rá✈❡✐s✳ ❖s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ q✉❡ ♥ã♦ sã♦ ❛❧❣é❜r✐❝♦s✱ sã♦ ❝❤❛♠❛❞♦s tr❛♥s❝❡♥❞❡♥t❡s✳ P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s tr❛♥s❝❡♥❞❡♥t❡s é ♥ã♦ ❡♥✉♠❡rá✈❡❧ ❡ ❡♠ ♣❛r✲ t✐❝✉❧❛r✱ é ♥ã♦ ✈❛③✐♦✳ ❆♣r❡s❡♥t❛♠♦s ♦ ❝❡❧❡❜r❛❞♦ ❚❡♦r❡♠❛ ❞❡ ▲✐♥❞❡♠❛♥♥✲❲❡✐❡rstr❛ss s❡♠ ♣r♦✈❛✱ ❞❛❞♦ ♦ ❝❛rát❡r té❝♥✐❝♦ ❞❛ ♣r♦✈❛✱ ♠❛s ✉s❛♠♦s ❡❧❡ ♣❛r❛ ♣r♦✈❛r ❝♦♠♦ ❝♦r♦✲ ❧ár✐♦ q✉❡ ♦s ♥ú♠❡r♦s e ❡ π sã♦ tr❛♥s❝❡♥❞❡♥t❡s✳
❋✐♥❛❧♠❡♥t❡ ♥♦ ❝❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❜r❡✈❡ ❤✐stór✐❛ s♦❜r❡ π✱ ❛❧❣✉♠❛s
❢ór♠✉❧❛s ❡ sér✐❡s q✉❡ ♣❡r♠✐t❡♠ ❝❛❧❝✉❧á✲❧♦✱ ❛ss✐♠ ❝♦♠♦ ❛♣r♦①✐♠❛çõ❡s ✉s❛♥❞♦ ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛✳ ❖ tr❛❜❛❧❤♦ ❛❝❛❜❛ ❝♦♠ ❛❧❣✉♠❛s ❝✉r✐♦s✐❞❛❞❡s s♦❜r❡ ❡st❡ ♥ú♠❡r♦ ♠❛r❛✈✐❧❤♦s♦✱ q✉❡ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦s ♥ú♠❡r♦se❀i❀1❀0sã♦✱ ❛♦ ❞✐③❡r ❞❡ ❊✉❧❡r✱ ♦s ♠❛✐s
✐♠♣♦rt❛♥t❡s ❞❛ ♠❛t❡♠át✐❝❛ ❡ q✉❡ ❡❧❡ ♦s ❥✉♥t♦✉ ♥❛ s✉❛ ❢❛♠♦s❛ ❢ór♠✉❧❛✿ eiπ+ 1 = 0✳ ❋❛③❡♥❞♦ ♠❡♥çã♦ ❛♦ ✜❧♠❡ ✐♥❞✐❝❛❞♦ ❛♦ ❖❙❈❆❘✱ r❡s✉♠✐♠♦s ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♠♦ ❚❤❡ ▲✐❢❡ ♦❢ π✳
❈❛♣ít✉❧♦ ✶
❉❡✜♥✐çã♦ ❡ ❡①✐stê♥❝✐❛ ❞♦ ♥ú♠❡r♦
π
◆❡st❡ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ✉s❛r❡♠♦s ❛ ❝♦♠♣❧❡t✉❞❡ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣❛r❛ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞♦ ♥ú♠❡r♦ π✳
✶✳✶ ❆ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦
❈♦♥s✐❞❡r❡ ✉♠ ❛r❝♦AB⌢ ❞❡ ✉♠ ❝ír❝✉❧♦ ❈✿
❚♦♠❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s A0, A1, A2, . . . , An s♦❜r❡ ♦ ❛r❝♦✱ t❛✐s q✉❡
A0 =A ❡ An=B✱ ❡ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s ❝♦♥s❡❝✉t✐✈♦s Ai−1, Ai✱ ❝♦♥s✐❞❡r❡♠♦s ♦
s❡❣♠❡♥t♦ Ai−1Ai✳ ❆ ✉♥✐ã♦ ❞❡ss❡s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❣❡r❛ ✉♠❛ ♣♦❧✐❣♦♥❛❧ ♣♦♥t✐❧❤❛❞❛
❡ ❛ s♦♠❛ ❞❡ s❡✉s ❝♦♠♣r✐♠❡♥t♦s é ❞❡♥♦t❛❞❛ ♣♦r pn✱ ♦✉ s❡❥❛✱
pn =A0A1+A1A2+· · ·+An−1An = n X
i=1
Ai−1Ai,
♦♥❞❡ Ai−1Ai ❞❡♥♦t❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ Ai−1Ai✳
✶✳✶✳ ❆ ❉❊❋■◆■➬➹❖ ❉❊ ❈❖▼P❘■▼❊◆❚❖ ❉❊ ❆❘❈❖
❙❡❥❛ P ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♥ú♠❡r♦s pn q✉❡ sã♦ ❝♦♠♣r✐♠❡♥t♦s ❞❡ ❧✐♥❤❛s ♣♦♥t✐❧❤❛❞❛s ✐♥s❝r✐t❛s ❡♠ AB⌢ ✱ ❝♦♠♦ ❛❝✐♠❛✱ ♦✉ s❡❥❛✱
P =
(
pn | pn = n X
i=1
Ai−1Ai )
◆♦t❡♠♦s q✉❡ P 6=∅✱ ♣♦✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AB é ✉♠
❡❧❡♠❡♥t♦ ❞❡ P✳
❆✜r♠❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦P é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡rá
s✉♣r❡♠♦✳ P❛r❛ ❥✉st✐✜❝❛r♠♦s ❡ss❛ ❛✜r♠❛çã♦✱ ♣r❡❝✐s❛r❡♠♦s ❞♦ s❡❣✉✐♥t❡✿
▲❡♠❛ ✶✳✶✳✶ ❙❡❥❛ △P QR ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✱ ❞❡ ❜❛s❡ QR✳ ❈♦♥s✐❞❡r❡♠♦s ♦s
♣♦♥t♦s ❙✱ ❯ ❡ ❚ ✭❝♦♠♦ ♥❛ ✜❣✉r❛✮ ❞❡ ♠♦❞♦ q✉❡ QR kSU✳ ❊♥tã♦ ST > QR✳
Pr♦✈❛✿ ❖❜s❡r✈❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ △P QR é s❡♠❡❧❤❛♥t❡ ❛♦ tr✐â♥❣✉❧♦ △P SU✳
▲♦❣♦
SU QR =
P S QP >1
❡✱ ♣♦rt❛♥t♦✱
SU > QR
❆❧é♠ ❞✐ss♦✱ ♦ â♥❣✉❧♦QRP[ é ❛❣✉❞♦✱ ♣♦✐s é ❜❛s❡ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✳ ❆ss✐♠✱
♦ â♥❣✉❧♦ QRU[ é ♦❜t✉s♦ ❡ ♦ â♥❣✉❧♦ RU S[ ❛❣✉❞♦✳ ❊♥tã♦✱ ♦ â♥❣✉❧♦ SU T[ é ♦❜t✉s♦ ❡ ♦
✶✳✶✳ ❆ ❉❊❋■◆■➬➹❖ ❉❊ ❈❖▼P❘■▼❊◆❚❖ ❉❊ ❆❘❈❖
â♥❣✉❧♦ ST U[ é ❛❣✉❞♦✳ ▼❛s✱ s❛❜❡♠♦s q✉❡✱ ♥✉♠ tr✐â♥❣✉❧♦✱ ♦ ❧❛❞♦ ❞❡ ♠❛✐♦r ♠❡❞✐❞❛ é
♦♣♦st♦ ❛♦ ♠❛✐♦r â♥❣✉❧♦✳ P♦rt❛♥t♦✱
ST > QR.
❆❣♦r❛✱ ♣r♦✈❡♠♦s ❛ ❛✜r♠❛çã♦ ❛♥t❡r✐♦r✳
❚❡♦r❡♠❛ ✶✳✶✳✶ ❖ ❝♦♥❥✉♥t♦ P é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✳
Pr♦✈❛✿ ❚♦♠❡♠♦s ✉♠ q✉❛❞r❛❞♦ q✉❡ ❝♦♥t❡♥❤❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ❡♠ s✉❛ t♦t❛❧✐❞❛❞❡✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ♣♦♥t♦sA′
i s♦❜r❡ ♦ q✉❛❞r❛❞♦✱ q✉❡ sã♦ ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞♦s ♣r♦❧♦♥❣❛♠❡♥t♦s ❞♦s r❡s♣❡❝t✐✈♦s s❡❣♠❡♥t♦s DAi✳
❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ♦ ▲❡♠❛ ✶✳✶✳✶ ✱ ✈❡♠♦s q✉❡ Ai−1Ai < A′i−1A′i✳ ▲♦❣♦✱
pn= n X
i=1
Ai−1Ai < n X
i=1
A′i−1A′i,
♠❛s n
X
i=1
A′
i−1A′i ≤q,
♦♥❞❡ q é ♦ ♣❡rí♠❡tr♦ ❞♦ q✉❛❞r❛❞♦✳ ❆ss✐♠✱ pn ≤ q ♣❛r❛ t♦❞♦ n ∈ N✳ ■st♦ ♣r♦✈❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ P é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✳
❖❜s❡r✈❛çã♦✿ P❡❧♦ ❛①✐♦♠❛ ❞❛ ❝♦♠♣❧❡t✉❞❡ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♦ ❝♦♥❥✉♥t♦ P ♣♦ss✉✐
s✉♣r❡♠♦ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r ♣✳ ■st♦ ♥♦s ♣❡r♠✐t❡ ❞❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
✶✳✷✳ ❆ ❊❳■❙❚✃◆❈■❆ ❉❊ π
❉❡✜♥✐çã♦ ✶ ❉❡✜♥✐♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ AB ❝♦♠♦ s❡♥❞♦ ♦ s✉♣r❡♠♦ ❞♦
❝♦♥❥✉♥t♦ P✱ ✐st♦ é✱
ℓ
⌢
(AB):=p=supP
❖❜s❡r✈❛çã♦✿ P❛r❛ ❞❡✜♥✐r♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈✱ ♣r♦❝❡❞❡♠♦s ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳ P❛r❛ ✐st♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ❞❡ ✈ér✲ t✐❝❡s
A0, A1, A2, . . . , An−1, An=A0.
❈♦♥s✐❞❡r❡♠♦spn ❛ s♦♠❛ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞❛s ❧✐♥❤❛s ♣♦♥t✐❧❤❛❞❛s ✐♥s❝r✐t❛s✱ ✐st♦ é✱
pn = n X
i=1
Ai−1Ai,
❙❡❥❛ P ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♠♣r✐♠❡♥t♦s pn✳ Pr♦✈❛✲s❡✱ ❝♦♠♦ ❛❝✐♠❛✱ q✉❡ P 6=∅❡ q✉❡ P é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✳ P❡❧♦ ❛①✐♦♠❛ ❞❛ ❝♦♠♣❧❡t✉❞❡ ❞♦s ♥ú♠❡r♦s
r❡❛✐s✱ ❡①✐st❡ p= supP✳ ■st♦ ♥♦s ♣❡r♠✐t❡ ♥♦s ❞❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✷ ❉❡✜♥✐♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ♣♦r
p=ℓ(C) := supP
✶✳✷ ❆ ❡①✐stê♥❝✐❛ ❞❡
π
◆❡st❛ s❡çã♦✱ ♣r♦✈❛r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞♦ ♥ú♠❡r♦ π✳ ▼♦str❛r❡♠♦s q✉❡ ❛ r❛③ã♦
❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ q✉❛❧q✉❡r ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ s❡✉ ❞✐â♠❡tr♦ é ❝♦♥st❛♥t❡✳ ❊ss❛ r❛③ã♦ s❡rá ❞❡♥♦t❛❞❛ ♣♦r π✳ ❆♥t❡s ❞✐ss♦✱ ♣r❡❝✐s❛r❡♠♦s ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s✿
❙❡❥❛ P ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ ❡ ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ t❛❧ q✉❡ P ⊂ R≥0 ❡ s❡❥❛
k ∈R+✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦✿
kP :={kp|p∈P}
P♦r ❡①❡♠♣❧♦✱ s❡
P = [0,1] ❡k = 3
❡♥tã♦✱
3P = [0,3].
❖❜s❡r✈❡♠♦s q✉❡ s❡ j ∈R+✱ ❡♥tã♦
j(kP) = (jk)P.
✶✳✷✳ ❆ ❊❳■❙❚✃◆❈■❆ ❉❊ π
▲❡♠❛ ✶✳✷✳✶ ❙❡ b é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ P✱ ❡♥tã♦ kb é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ kP✳
Pr♦✈❛✿ ❙❡ b é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ P✱ ❡♥tã♦
p ≤ b ♣❛r❛ t♦❞♦p∈P,
❡ ❝♦♠♦ k > 0✱ ❡♥tã♦ kb≤kp ♣❛r❛ t♦❞♦ p∈P.
P♦rt❛♥t♦✱ kb é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ kP.
◆♦t❡♠♦s q✉❡ ♦ ▲❡♠❛ ❛♥t❡r✐♦r ♠♦str❛ q✉❡ kP é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡
▲❡♠❛ ✶✳✷✳✷ ❙❡ ❝ é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ kP✱ ❡♥tã♦ c/k é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ P✳
Pr♦✈❛✿ ❙❡ ❝ é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ kP✱ ❡♥tã♦
kp≤c, ♣❛r❛ t♦❞♦ p∈P.
❈♦♠♦ k > 0✱ t❡♠♦s q✉❡
p≤ c
k, ♣❛r❛ t♦❞♦ p∈P,
♦ q✉❡ ❣❛r❛♥t❡ q✉❡ c
k é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ P✳
❈♦♠♦kP 6=∅✭♣♦✐sP 6=∅✮ ❡kP é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✱ ♦ ❛①✐♦♠❛ ❞❛ ❝♦♠♣❧❡t✉❞❡
❞♦s ♥ú♠❡r♦s r❡❛✐s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ sup(kP)✳ ▼❛✐s ❛✐♥❞❛✱ t❡♠♦s ❛ s❡❣✉✐♥t❡
r❡❧❛çã♦✿
▲❡♠❛ ✶✳✷✳✸ sup(kP) =ksup(P) ♣❛r❛ t♦❞♦k ≥0
Pr♦✈❛✿ ❙❡❥❛ ❜ ♦ s✉♣r❡♠♦ ❞❡ P✳ ❊♥tã♦✱ ❜ é ❝♦t❛ s✉♣❡r✐♦r ❞❡ P✳ P❡❧♦ ▲❡♠❛ ✶✳✷✳✶✱ ❦❜ é ❝♦t❛ s✉♣❡r✐♦r ❞❡kP✳ ❆✜r♠❛♠♦s q✉❡kb é ♦ s✉♣r❡♠♦ ❞❡ kP✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s✱
♣♦r ❛❜s✉r❞♦✱ q✉❡ kP t❡♠ ❝♦t❛ ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❝ t❛❧ q✉❡
c < kb.
❉♦ ❧❡♠❛ ✶✳✷✳✷ ✱ t❡♠♦s q✉❡c/k é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ P q✉❡ s❡r✐❛ ♠❡♥♦r q✉❡ ❜✱ ♦ q✉❡
é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s ❜ é ❛ ✏♠❡♥♦r✑ ❞❛s ❝♦t❛s s✉♣❡r✐♦r❡s ❞❡ P✳
❆❣♦r❛ ❡st❛♠♦s ♣r♦♥t♦s ♣❛r❛ ♣r♦✈❛r q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❡ ♦ ❞✐â♠❡tr♦ ❞❡ t♦❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ❝♦♥st❛♥t❡✳
✶✳✷✳ ❆ ❊❳■❙❚✃◆❈■❆ ❉❊ π
❚❡♦r❡♠❛ ✶✳✷✳✶ ✭❡①✐stê♥❝✐❛ ❞❡ π✮ ❙❡❥❛♠ C ❡ C′ ❞♦✐s ❝ír❝✉❧♦s ❞❡ r❛✐♦s r ❡ r′ ❡ ❝♦♠♣r✐♠❡♥t♦s p ❡ p′✱ ❡♥tã♦✱
p
2r = p′
2r′.
Pr♦✈❛✿ ❙✉♣♦♥❤❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ ♦s ❝ír❝✉❧♦s sã♦ ❝♦♥❝ê♥tr✐❝♦s ❞❡ ❝❡♥tr♦ ❉ ❡ q✉❡ r′ > r✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✿
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s A0, A1, A2, . . . , An−1, An = A0 s♦❜r❡ ♦ ❝ír✲
❝✉❧♦ ❈✳ P❛r❛ ❝❛❞❛0≤i≤n✱ s❡❥❛A′
i ♦ ♣♦♥t♦ ❞❡ ❈✬ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦DAi✳ ❆ss✐♠✱ ♦❜t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦sA′
0, A′1, A′2, . . . , A′n=A′0
s♦❜r❡ ♦ ❝ír❝✉❧♦ ❈✬✳ ▲♦❣♦✱ ♦s tr✐â♥❣✉❧♦s
∆DAi−1Ai ∼∆DA′i−1A′i. sã♦ s❡♠❡❧❤❛♥t❡s✳ P♦rt❛♥t♦✱
A′ i−1A′i
Ai−1Ai
= r
′
r
❆❣♦r❛✱ s❡ ♦s ♣❡rí♠❡tr♦s ❞♦s ♣♦❧í❣♦♥♦s✱ ✐♥s❝r✐t♦s ❡♠ C ❡ C′✱ ❢♦r♠❛❞♦s ♣❡❧♦s s❡❣✲ ♠❡♥t♦s Ai−1Ai ❡ A′i−1A′i sã♦ ❞❡♥♦t❛❞♦s ♣♦r pn ❡ p′n✱ t❡♠♦s q✉❡
p′ n=
n X
i=1
A′
i−1A′i = n X
i=1
r′
r
(Ai−1Ai) =
r′
r
n X
i=1
(Ai−1Ai) =
r′
rpn.
❆ss✐♠✱
p′ n=
r′
rpn, ♣❛r❛ t♦❞♦ n ∈N ✭✶✳✶✮
❖❜s❡r✈❡♠♦s q✉❡ ♦s ♣♦♥t♦s A′
i ❢♦r❛♠ ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞♦s ♣♦♥t♦s Ai✱ ♠❛s ♦ ♣r♦✲ ❝❡ss♦ é r❡✈❡rsí✈❡❧✱ ✐st♦ é✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♣r✐♠❡✐r♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s
✶✳✷✳ ❆ ❊❳■❙❚✃◆❈■❆ ❉❊ π
A′
0, A′1, A′2, . . . , A′n = A′0 s♦❜r❡ ❈✬ ❡ ♦❜t❡r ♦s ♣♦♥t♦s A0, A1, A2, . . . , An−1, An = A0
s♦❜r❡ ❈ ❝♦♠♦ ✐♥t❡rs❡❝çõ❡s ❞♦s s❡❣♠❡♥t♦s DA′
i ❝♦♠ ♦ ❝ír❝✉❧♦ ❈✳ P❡❧❛ ❉❡✜♥✐çã♦ ✷✱
p= supP = sup
(
pn | pn = n X
i=1
Ai−1Ai )
❡
p′ = supP′ = sup (
p′
n | p′n= n X
i=1
A′ i−1A′i
)
❉❛ ❡①♣r❡ssã♦ ✭✶✳✶✮ ❡ ❞♦ ❧❡♠❛ ✶✳✷✳✸ ❢❛③❡♥❞♦ k = r
′
r
✱ t❡♠♦s q✉❡
P′ = r′
r ·P.
P♦rt❛♥t♦
supP′ = r′
r supP, ♦✉ s❡❥❛✱ p′
r′ =
p r.
❉✐✈✐❞✐♥❞♦ ♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ♣♦r ✷✱ t❡♠♦s ♥♦ss♦ ❡♥✉♥❝✐❛❞♦✳
❖✉tr♦ t❡♦r❡♠❛ ❛♥á❧♦❣♦ ❛♦ ❛♥t❡r✐♦r ♣♦❞❡ s❡r ♣r♦✈❛❞♦ ♣❛r❛ ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛✳
❚❡♦r❡♠❛ ✶✳✷✳✷ ❙❡❥❛♠ ❈ ❡ ❈✬ ❞♦✐s ❝ír❝✉❧♦s ❞❡ r❛✐♦ r ❡ r✬✱r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ ⌢
AB ❡
⌢
A′B′ ❛r❝♦s ❞❡ ❈ ❡ ❈✬✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛ ❡♠ ❣r❛✉s✳ ❙❡ ♣ ❡ ♣✬ sã♦ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❛r❝♦s AB⌢ ❡ A⌢′B′✱ ❡♥tã♦
p r =
p′
r′
✶✳✸✳ ❈➪▲❈❯▲❖ ❉❖ ❈❖▼P❘■▼❊◆❚❖ ❉❖ ❆❘❈❖ ❯❙❆◆❉❖ ▲■▼■❚❊❙
❖❜s❡r✈❛çõ❡s✿
✭✶✮ ❊st❛ r❛③ã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♠❡❞✐❞❛ ❡♠ r❛❞✐❛♥♦s ❞♦ ∠BCA✳ ❖ t❡♦r❡♠❛
❞✐③ q✉❡ ❛ ♠❡❞✐❞❛ ❡♠ r❛❞✐❛♥♦s ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞♦ â♥❣✉❧♦✱ ♦✉ q✉❡ ❛ ♠❡❞✐❞❛ ❡♠ ❣r❛✉s ❞♦ ❛r❝♦✱ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦✳ ❙❡ θ ❞❡♥♦t❛ ❛ ♠❡❞✐❞❛
❡♠ r❛❞✐❛♥♦s ❞♦ â♥❣✉❧♦ ∠BCA✱ t❡♠♦s q✉❡ θ = p
r✱ ❞♦♥❞❡ p = θr ✱ ❛ ❝❧áss✐❝❛
❢ór♠✉❧❛ ❞❛ tr✐❣♦♥♦♠❡tr✐❛✳
✭✷✮ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ é s❡♠❡❧❤❛♥t❡ à ❞♦ t❡♦r❡♠❛ ✶✳✷✳✶✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ♦s ❝ír❝✉❧♦s sã♦ ❝♦♥❝ê♥tr✐❝♦s ❡ t♦♠❛r s❡q✉ê♥✲ ❝✐❛s ❞❡ ♣♦♥t♦s A = A1, A2, . . . , An = B ❡ A = A′1, A′2, . . . , A′n = B′ ❝♦♠♦ ♥❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳
✶✳✸ ❈á❧❝✉❧♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ ✉s❛♥❞♦ ❧✐♠✐t❡s
◆❡st❛ s❡çã♦✱ ✈❡r❡♠♦s q✉❡ ♦s ♣❡rí♠❡tr♦s ❞❛s ❧✐♥❤❛s ♣♦♥t✐❧❤❛❞❛s ✐♥s❝r✐t❛s ❞❡ ✉♠ ❛r❝♦ s❡ ❛♣r♦①✐♠❛♠ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ q✉❛♥❞♦ ❛s ♠❡❞✐❞❛s ❞♦ ❧❛❞♦ ♠❛✐♦r ❞❛s ❧✐♥❤❛s ♣♦♥t✐❧❤❛❞❛s s❡ ❛♣r♦①✐♠❛♠ ❞❡ ③❡r♦✳
❆♥t❡s ❞❡ t♦r♥❛r ♠❛✐s ♣r❡❝✐s❛ ♥♦ss❛ ❛✜r♠❛çã♦✱ ❝♦♥s✐❞❡r❡♠♦s AB⌢ ✉♠ ❛r❝♦ s♦❜r❡
✉♠ ❝ír❝✉❧♦✱ ❝♦♠♦ ♥❛ s❡❣✉✐♥t❡ ✜❣✉r❛✿
P❛r❛ ❝❛❞❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦sA=A1, A2, . . . , An=Bs♦❜r❡ ♦ ❝ír❝✉❧♦ ⌢
AB✱ ❞❡♥♦✲
t❡♠♦s ❝♦♠♦Bn❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ q✉❡ é ❛ ✉♥✐ã♦ ❞♦s s❡❣♠❡♥t♦sAi−1, Ai✱1≤ i≤ n
❡ ❞❡♥♦t❡♠♦s ❝♦♠ m(Bn) ♦ ♠❛✐♦r ❞♦s ❝♦♠♣r✐♠❡♥t♦s✱ ✐st♦ é✱
m(Bn) := max{Ai−1, Ai | 1≤i≤n}.
✶✳✸✳ ❈➪▲❈❯▲❖ ❉❖ ❈❖▼P❘■▼❊◆❚❖ ❉❖ ❆❘❈❖ ❯❙❆◆❉❖ ▲■▼■❚❊❙
➱ ❝❧❛r♦ q✉❡ m(Bn) ≤ pn✱ ♦♥❞❡ pn = n X
i=1
Ai−1, Ai. ❈♦♠♦ ❛♥t❡s✱ s❡❥❛
P =
(
pn | pn= n X
i=1
Ai−1, Ai )
. P♦r ❞❡✜♥✐çã♦✱
ℓ(AB⌢ ) = p✱ ♦♥❞❡ p= supP
❆❣♦r❛✱ ♣r❡❝✐s❛♠♦s ♥♦ss❛ ❛✜r♠❛çã♦ ❝♦♠ ♦ s❡❣✉✐♥t❡✿
❚❡♦r❡♠❛ ✶✳✸✳✶ ❙❡❥❛ AB⌢ ✉♠ ❛r❝♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ p = supP✳ P❛r❛ t♦❞♦ ε > 0
❡①✐st❡ δ >0 t❛❧ q✉❡ s❡ m(Bn)< δ✱ ❡♥tã♦ pn > p−ε✳ Pr♦✈❛✿ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ s✉♣r❡♠♦✱ ❡①✐st❡ p′
n ∈P t❛❧ q✉❡
p′n> p−
ε
2.
❙❡❥❛ B′
n ✉♠❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✐♥s❝r✐t❛ ♥♦ ❝ír❝✉❧♦ ❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ p′n.
❆❣♦r❛✱ s❡❥❛ Bm ✉♠❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✐♥s❝r✐t❛ ♥✉♠ ❝ír❝✉❧♦ q✉❛❧q✉❡r ❞❡ ❝♦♠♣r✐✲ ♠❡♥t♦ Pm✳ ❙❡❥❛ B′′
r ❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ♦❜t✐❞❛ ✉s❛♥❞♦ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡ Bm ❡ ❞❡
Bn✳ ❙❡❥❛′ p
r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ Br′′✳
P❛r❛ ✐❧✉str❛r ✐st♦✱ ♥❛ ✜❣✉r❛✱ ♦s ♣♦♥t♦s ♠❛r❝❛❞♦s ❝♦♠ ❝r✉③❡s sã♦ ♦s ✈ért✐❝❡s ❞❡
B′
n ❡ ♦s ♣♦♥t♦s ♠❛r❝❛❞♦s ❝♦♠ ❝ír❝✉❧♦s✱ sã♦ ♦s ✈ért✐❝❡s ❞❡ Bm✳
❆♣ós r❡♣❡t✐❞❛s ❛♣❧✐❝❛çõ❡s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s q✉❡ p′′
r ≥ p′n✱ ♣♦rt❛♥t♦
p′′
r > p−
ε
2.
❚❛♠❜é♠ é ✐♠❡❞✐❛t♦ ♣❡r❝❡❜❡r q✉❡pm ≤p′′r✱ ♣♦✐s ❝❛❞❛ ✈❡③ q✉❡ t♦♠❛r♠♦s ✉♠ ✈ért✐❝❡ ❞❡ B′′
r✱ ♣r♦✈❡♥✐❡♥t❡ ❛♣❡♥❛s ❞❛ ♠❛r❝❛çã♦ ❞❡ Bn✱ t❡r❡♠♦s ✉♠❛ r❡❞✉çã♦ ♥❛ ❢♦r♠❛çã♦′ ❞❡ Bm ❡♠ r❡❧❛çã♦ à ❞❡Br′′✳ ❊ss❛ ✏r❡❞✉çã♦✑ ✜❝❛ ❛✐♥❞❛ ♠❛✐s ❝❧❛r❛ q✉❛♥❞♦ ❛♥❛❧✐s❛♠♦s ❛ ✜❣✉r❛ ❛ s❡❣✉✐r✿
✶✳✸✳ ❈➪▲❈❯▲❖ ❉❖ ❈❖▼P❘■▼❊◆❚❖ ❉❖ ❆❘❈❖ ❯❙❆◆❉❖ ▲■▼■❚❊❙
◆♦t❡ q✉❡P R < P Q+P Q✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮✱ ♦♥❞❡P R∈Bm ❡P Q, P R∈ B′n✳ ❖ ♥ú♠❡r♦ ❞❡ ✈❡③❡s ❡♠ q✉❡ ❡ss❛ r❡❞✉çã♦ ♦❝♦rr❡ é✱ ♥♦ ♠á①✐♠♦✱ n−1 ✈❡③❡s ♣❛r❛ ♦
❝❛s♦ ❡♠ q✉❡ B′
n t❡♠ ♠❛r❝❛çõ❡s ❛♣❡♥❛s ❡♠ ❆ ❡ ❇✱ ♣♦❞❡♥❞♦ ❛✐♥❞❛✱ ♥ã♦ ♦❝♦rr❡r ✈❡③ ❛❧❣✉♠❛✱ ❝❛s♦ B′
n t❡♥❤❛ ♠❛r❝❛çõ❡s s♦❜r❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ Bm✳ ❙❡ ❞❡♥♦t❛♠♦s ❝♦♠
k ♦ ♠❛✐♦r ❧❛❞♦ ❞❡ Bm✱ ❡♥tã♦
p′′r−pm ≤(n−1).k P♦r ♦✉tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡ P′′
r > p−
ε
2✱ ✐st♦ é✱ p−p′′r <
ε
2✳
▲♦❣♦✱
p−pm = (p−pr′′) + (p′′r−pm)<
ε
2 + (n−1)k.
❈♦♠♦ ❞❡s❡❥❛♠♦s ❝♦♥❝❧✉✐r q✉❡ p−pm < ε✱ ✐st♦ s❡❣✉✐rá s❡ (n−1)k <
ε
2✱ ♦✉ s❡❥❛✱
k < ε
2(n−1).
❊♥tã♦✱ ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡stá r❡s♦❧✈✐❞♦ s❡ ❡s❝♦❧❤❡r♠♦s δ ≤ ε
2(n−1)✱ ♣♦✐s s❡ k < δ✱
❡♥tã♦ p−pm < ε ❡ ❝♦♠♦ pm≤p✱ ✐st♦ s❡rá ❡q✉✐✈❛❧❡♥t❡ ❛ |p−pm|< ε.
❖❜s❡r✈❛çã♦✿ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ t❡♦r❡♠❛ ❞✐③ q✉❡ pn s❡ ❛♣r♦①✐♠❛ ❞❡ p q✉❛♥❞♦ ♦ ♠❛✐♦r ❧❛❞♦ ❞❡ Bn✱ s❡ ❛♣r♦①✐♠❛ ❞❡ ③❡r♦✳ ■st♦✱ ♦❜✈✐❛♠❡♥t❡✱ ❡q✉✐✈❛❧❡ à ❛♣❧✐❝❛çã♦ ❞❡
✉♠ ❧✐♠✐t❡✳
✶✳✹✳ ❚❊❖❘❊▼❆ ❉❆ ❆❉■➬➹❖ P❆❘❆ ❈❖▼P❘■▼❊◆❚❖ ❉❊ ❆❘❈❖
✶✳✹ ❚❡♦r❡♠❛ ❞❛ ❛❞✐çã♦ ♣❛r❛ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦
❯♠ ❞♦s ♣♦st✉❧❛❞♦s ♣❛r❛ ♠❡❞✐❞❛ ❛♥❣✉❧❛r é ♦ ♣♦st✉❧❛❞♦ ❞❛ ❛❞✐çã♦✳ ❊st❡ ❛✜r♠❛ q✉❡ s❡ ✉♠ ♣♦♥t♦C ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦∠DAB✱ ❡♥tã♦m∠DAB =m∠DAC+
m∠CAB.
❆ ♣❛rt✐r ❞✐st♦✱ ♣♦❞❡✲s❡ ♣r♦✈❛r ✉♠ r❡s✉❧t❛❞♦ s❡♠❡❧❤❛♥t❡ ♣❛r❛ ❛ ♠❡❞✐❞❛ ❛♥❣✉❧❛r ❞❡ ❛r❝♦s ❝✐r❝✉❧❛r❡s✳ ■st♦ é✱ m ABC⌢ = m AB⌢ +m BC⌢ ✳ ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ é ♦
❡♥✉♥❝✐❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ ❝♦♠♣r✐♠❡♥t♦s ❞❡ ❛r❝♦✳
❚❡♦r❡♠❛ ✶✳✹✳✶ ❙❡❥❛♠ AB⌢ ❡ BC⌢ ❛r❝♦s ❞❡ ✉♠ ♠❡s♠♦ ❝ír❝✉❧♦✱ ❝♦♠ s♦♠❡♥t❡ ✉♠
♣♦♥t♦ ❇ ❡♠ ❝♦♠✉♠ ♣❛r❛ ❛♠❜♦s✳ ❙❡❥❛♠ s1 ❡ s2 ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ AB ❡ BC✱ ❡
s❡❥❛ s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ABC✳ ❊♥tã♦✱
s1+s2 =s
Pr♦✈❛✿ ✭✶✮ ❙✉♣♦♥❤❛♠♦s q✉❡ s1+s2 > s✳ ❊♥tã♦✱ s1+s2−s >0✳ ❙❡❥❛
ε=s1+s2−s.
✶✳✹✳ ❚❊❖❘❊▼❆ ❉❆ ❆❉■➬➹❖ P❆❘❆ ❈❖▼P❘■▼❊◆❚❖ ❉❊ ❆❘❈❖
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ Bn ✐♥s❝r✐t❛ ❡♠ ⌢
AB ❞❡ ❝♦♠♣r✐♠❡♥t♦ pn✱ ❞❡
♠♦❞♦ q✉❡
pn > s1−
ε
2.
❆♥❛❧♦❣❛♠❡♥t❡✱ s❡❥❛ B′
m ✉♠❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✐♥s❝r✐t❛ ❡♠ ⌢
BC✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ p′m✱ ❞❡ ♠♦❞♦ q✉❡
p′m > s2−
ε
2.
❋❛③❡♥❞♦ ❛ ✉♥✐ã♦ ❞❡ss❛s ❧✐♥❤❛s ♣♦♥t✐❧❤❛❞❛s✱ ♦❜t❡♠♦s ✉♠❛ ♥♦✈❛ ❧✐♥❤❛ ❞❡♥♦t❛❞❛ ♣♦r
B′′
m+n ❞❡ ❝♦♠♣r✐♠❡♥t♦ p′′m+n t❛❧ q✉❡
p′′
m+n =pn+p′m ❈♦♠♦ B′′
m+n ❡stá ✐♥s❝r✐t♦ ❡♠ ⌢
ABC✱ s❡❣✉❡✲s❡ q✉❡ p′′
n+m ≤s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡
p′′
m+n =pn+pm′ > s1+s2−ε=s
❉❡ss❛ ❢♦r♠❛✱ p′′
m+n≤s ❡ p′′m+n> s ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳
✭✷✮ ❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ s1+s2 < s✳ ❊♥tã♦ s−s1−s2 >0✳ ❙❡❥❛ ε=s−s1−s2✳
❆ss✐♠✱
s=s1+s2+ε.
❙❡❥❛ δ ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ❞❡ ♠♦❞♦ q✉❡ s❡ Bn é ✉♠❛ ❧✐♥❤❛ ✐♥s❝r✐t❛ ❡♠ ⌢
AB ❞❡
❝♦♠♣r✐♠❡♥t♦ pn ❡ m(Bn)< δ✱ ❡♥tã♦
s−pn< ε ou pn > s−ε.
❚❛❧ δ ❡①✐st❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✳✶✳ ❆❣♦r❛✱ s❡❥❛ Bn ✉♠❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✐♥s❝r✐t❛ ❡♠ ⌢
ABC ❝♦♠ m(bn) < δ t❛❧ q✉❡ ❇ é ✉♠ ✈ért✐❝❡ ❞❡ Bn✳ ❙❡❥❛ pn ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡
Bn✱ ❡♥tã♦ Bn ♣♦❞❡ s❡r ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ❧✐♥❤❛s ♣♦♥t✐❧❤❛❞❛s Bm′ , Br′ (m +r = n)✱ ✉♠❛ ✐♥s❝r✐t❛ ❡♠ AB⌢ ❡ ❛ ♦✉tr❛ ✐♥s❝r✐t❛ ❡♠ BC⌢ ✳ ❙❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ss❛s ❧✐♥❤❛s
♣♦♥t✐❧❤❛❞❛s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ p′
m ❡ p′′r✱ t❡♠♦s q✉❡
p′m+p′′r ≤pn,
p′
m≤s1,
p′′ r ≤s2.
P♦rt❛♥t♦✱
pn ≤s1+s2
✶✳✺✳ ❆P❘❖❳■▼❆◆❉❖ ❆ ➪❘❊❆ ❉❊ ❯▼ ❙❊❚❖❘ ❈■❘❈❯▲❆❘ P❖❘ ❉❊❋❊■❚❖
▼❛s✱ s1+s2 =s−ε✳ ❊♥tã♦
pn≤s−ε,
♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱s1+s2 =s✳
✶✳✺ ❆♣r♦①✐♠❛♥❞♦ ❛ ár❡❛ ❞❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r ♣♦r
❞❡❢❡✐t♦
❙❡❥❛ ❈ ✉♠ ❝ír❝✉❧♦ ❝♦♠ ❝❡♥tr♦ ❈ ❡ r❛✐♦ r ❡ s❡❥❛♠ AB⌢ ✉♠ ❛r❝♦ ❞❡ ❈ ❡ ❑ ❛ ✉♥✐ã♦
❞❡ t♦❞♦s ♦s r❛✐♦s CP✱ ♦♥❞❡ p∈AB⌢ ✳
❉❡✜♥✐çã♦ ✸ ❖ ❝♦♥❥✉♥t♦ ❑ é ❝❤❛♠❛❞♦ s❡t♦r ❝✐r❝✉❧❛r ❞❡ r❛✐♦ r ❡ ❢r♦♥t❡✐r❛ AB⌢ ✳
❚❡♦r❡♠❛ ✶✳✺✳✶ ❙❡❥❛ K ✉♠ s❡t♦r ❝✐r❝✉❧❛r ❞❡ r❛✐♦ r ❡ ❢r♦♥t❡✐r❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ s✳
❊♥tã♦✱ ❤á ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐s K1, K2, . . . t♦❞❛s ❝♦♥t✐❞❛s ❡♠ K ❞❡
t❛❧ ♠♦❞♦ q✉❡ ❡①✐st❡
lim
n→∞αKn =
1 2 ·r·s,
♦♥❞❡ αKn ❞❡♥♦t❛ ❛ ár❡❛ ❞❡ Kn✳
Pr♦✈❛✿ ❙❡❥❛ Bn ✉♠❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ✐♥s❝r✐t❛ ♥♦ ❛r❝♦ AB ♥❛ q✉❛❧ t♦❞♦s ♦s s❡✉s ❧❛❞♦s sã♦ ❝♦♥❣r✉❡♥t❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦ bn✿
✶✳✺✳ ❆P❘❖❳■▼❆◆❉❖ ❆ ➪❘❊❆ ❉❊ ❯▼ ❙❊❚❖❘ ❈■❘❈❯▲❆❘ P❖❘ ❉❊❋❊■❚❖
❖❜s❡r✈❛♥❞♦ ❛ ✜❣✉r❛✱ é ❢á❝✐❧ ✈❡r q✉❡ ♦s tr✐â♥❣✉❧♦s ∆An−1AnC sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦r✲ t❛♥t♦✱ ❛s ❛❧t✉r❛s tê♠ ♠❡s♠❛ ♠❡❞✐❞❛✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r an✳ ❆ss✐♠✱ ❛ ár❡❛ ❞❡
q✉❛❧q✉❡r ✉♠ ❞❡ss❡s tr✐â♥❣✉❧♦s é 1
2anbn✳ ❙❡❥❛ ❛❣♦r❛Kn ❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ q✉❡ ❝♦rr❡s✲
♣♦♥❞❡ à ✉♥✐ã♦ ❞❡ss❡s tr✐â♥❣✉❧♦s✳ ❊♥tã♦
αKn=
1 2nanbn
❙❡❥❛s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡AB⌢ ✳ ◆♦t❡♠♦s q✉❡nbn é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛
Bn✳ ▲♦❣♦✱
nbn≤s ❡✱ ♣♦rt❛♥t♦✱
0≤bn≤
s n.
❈♦♠♦
lim
n→∞
s
n =s.nlim→∞
1
n =s.0 = 0,
s❡❣✉❡✲s❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❈♦♥❢r♦♥t♦ q✉❡✱
lim
n→∞bn = 0.
❈♦♠♦ m(Bn) = bn✱ ♣♦✐s t♦❞♦s ♦s s❡❣✉✐♠❡♥t♦s ❞❡ Bn sã♦ ✐❣✉❛✐s✱ s❡❣✉❡ q✉❡
lim
n→∞nbn=s
P♦r ♦✉tr♦ ❧❛❞♦✱ ❡①❛♠✐♥❛♥❞♦ ✉♠ tr✐â♥❣✉❧♦ ❞♦ t✐♣♦ ∆Ai−1CAi ✈❡♠♦s q✉❡
r < an−
bn
2.
P♦rt❛♥t♦✱
r− bn
2 < an < r.
✶✳✻✳ ❆P❘❖❳■▼❆◆❉❖ ❆ ➪❘❊❆ ❉❖ ❙❊❚❖❘ ❈■❘❈❯▲❆❘ P❖❘ ❊❳❈❊❙❙❖
❈♦♠♦ lim
n→∞bn= 0✱ ❡♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❈♦♥❢r♦♥t♦ t❡♠♦s q✉❡
lim
n→∞an=r ❉♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ t❡♠♦s q✉❡
lim
n→∞(αKn) = limn→∞
1 2.an.bn
= 1 2rs,
♦ q✉❡ ♣r♦✈❛ ♥♦ss♦ t❡♦r❡♠❛✳
✶✳✻ ❆♣r♦①✐♠❛♥❞♦ ❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ♣♦r ❡①✲
❝❡ss♦
◆❡st❛ s❡çã♦✱ ♠♦str❛r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐sL1, L2, . . .✱
❝❛❞❛ ✉♠❛ ❞❡❧❛s ❝♦♥t❡♥❞♦ ♦ s❡t♦r K ❞❡ ♠♦❞♦ q✉❡
lim
n→∞αLn=
1 2rs,
❡♠ q✉❡ αLn ❞❡♥♦t❛ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ Ln.
❚❡♦r❡♠❛ ✶✳✻✳✶ ❙❡❥❛♠ K ✉♠ s❡t♦r ❝✐r❝✉❧❛r ❞❡ r❛✐♦ r✱ ❞❡ ❢r♦♥t❡✐r❛ AB⌢ ❡ ❝♦♠♣r✐✲
♠❡♥t♦ s✱ ❡ s❡❥❛ ε ✉♠ ♥ú♠❡r♦ ❛r❜✐trár✐♦ ♣♦s✐t✐✈♦✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ L✱ ❝♦♥t❡♥❞♦ K✱ ❞❡ ♠♦❞♦ q✉❡
αL < 1
2rs+ε.
Pr♦✈❛✿ P❛r❛ ❡♥❝♦♥tr❛r t❛❧ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✱ tr❛❝❡♠♦s ✉♠ s❡t♦r ❝✐r❝✉❧❛r K′ ❝♦♠ ♦ ♠❡s♠♦ ❝❡♥tr♦ q✉❡ K ❡ r❛✐♦ r′ ❞❡ ♠♦❞♦ q✉❡ r′ > r✳
✶✳✻✳ ❆P❘❖❳■▼❆◆❉❖ ❆ ➪❘❊❆ ❉❖ ❙❊❚❖❘ ❈■❘❈❯▲❆❘ P❖❘ ❊❳❈❊❙❙❖
◆❛ ✜❣✉r❛✱ ❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ♥♦ s❡t♦r ♠❡♥♦r é ❛ ♠❡s♠❛ q✉❡ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ ❏á ❛ ❧✐♥❤❛ ✐♥s❝r✐t❛ ♥♦ s❡t♦r ♠❛✐♦r t❡♠ ❧❛❞♦s b′
n ❡ ❛❧t✉r❛a′n✳ ❙❡❥❛L ❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ ✐♥s❝r✐t❛ ♥♦ s❡t♦r ♠❛✐♦r✳ ❊♥tã♦
αL= n 2a
′ nb′n.
❙❡❥❛ s′ ❝♦♠♣r✐♠❡♥t♦ ❞❡A⌢′B′✳ P❡❧♦ t❡♦r❡♠❛ ✶✳✸✳✶✱ t❡♠♦s q✉❡
s′
r′ =
s r,
♣♦rt❛♥t♦
s′ = r
′.s
r
❆❧é♠ ❞✐ss♦✱ s❛❜❡♥❞♦ q✉❡
nb′ n ≤s′, t❡♠♦s
nb′ n≤
r′ ·s
r .
P♦r s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ s❡❣✉❡ q✉❡
a′ n
an
= r′
r,
♦✉ s❡❥❛✱
a′ n=
an·r′
r ❡ αL≤
1 2 ·
anr′
r · r′·s
r ≤
1 2
r′2·s
r
❆té ❡♥tã♦✱ t✉❞♦ é ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ r′ > r ❡ ♣❛r❛ t♦❞♦ n✳ Pr❡❝✐s❛♠♦s ❡s❝♦❧❤❡r r′ ❞❡ ♠♦❞♦ q✉❡ αL < 1
2rs+ε ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❡s❝♦❧❤❡r ♥✱ t❛❧ q✉❡ K ⊂L. ❉❡s❡❥❛♠♦s q✉❡
r′2s
r < rs+ 2ε, ♦✉ s❡❥❛✱ r
′2 < r2+ 2εr
s ,
♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱
r′ <pr2 + 2εr/s.
❊♥tã♦✱ ❜❛st❛ t♦♠❛rr′ > r✱ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❛♥t❡r✐♦r✳ ❉❡s❞❡ q✉❡ lim n→∞a
′ n=r′✱ s❡❣✉❡ q✉❡ a′
n > r ♣❛r❛ ❛❧❣✉♠ n. ❊s❝♦❧❤❡♥❞♦ ❡ss❡ n✱ t❡♠♦s q✉❡ L ❝♦♥té♠ K✳ ■st♦ ♣r♦✈❛ ♥♦ss♦ t❡♦r❡♠❛✳
❈❛♣ít✉❧♦ ✷
❉❡♠♦♥str❛çõ❡s ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡
❞❡
π
◆❡st❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ♠♦str❛❞❛s três ❞❡♠♦♥str❛çõ❡s ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡π✳ ■♥✐✲
❝✐❛❧♠❡♥t❡✱ ♠♦str❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s✱ q✉❡ ♣r♦✈❛rã♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳ ❊♠ s❡❣✉✐❞❛✱ ✉s❛♥❞♦ ❜❛s✐❝❛♠❡♥t❡ ❢❡rr❛♠❡♥t❛s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❢❛r❡♠♦s ❛ três ❞❡♠♦♥str❛çõ❡s✳
✷✳✶ ◆ú♠❡r♦s ♥❛t✉r❛✐s
❋❛③❡r r❡❢❡rê♥❝✐❛ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ✐♥❡✈✐t❛✈❡❧♠❡♥t❡✱ ❝♦rr❡s♣♦♥❞❡ ❛ ❛ss♦❝✐❛çõ❡s s♦❜r❡ ❝♦♥t❛❣❡♠✳ ❆ ✉t✐❧✐③❛çã♦ ❞❡ss❡s ♥ú♠❡r♦s é ❜❛st❛♥t❡ ❛♥t✐❣❛ ❡ ♠❡s♠♦ s❡♠ ♥♦çõ❡s ❢♦r♠❛✐s s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ✈ár✐❛s ❝✐✈✐❧✐③❛✲ çõ❡s✱ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ❞❡✐①❛r❛♠ r❡❣✐str♦s ❞♦ ✉s♦ ❞❡ss❡ ❝♦♥❥✉♥t♦✳
➱ ❞❡ ●✐✉ss❡♣❡ P❡❛♥♦ ✭✶✽✺✽ ✲ ✶✾✸✷✮ ❛ ❡❧❛❜♦r❛çã♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❯t✐❧✐③❛♥❞♦ ❛♣❡♥❛s q✉❛tr♦ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❛①✐♦♠❛s ❞❡ P❡❛♥♦✱ é ♣♦ssí✈❡❧ ♦❜t❡r t♦❞❛s ❛s ❛✜r♠❛çõ❡s ✈❡r❞❛❞❡✐r❛s q✉❡ ❝♦♥❤❡❝❡♠♦s s♦❜r❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
❍á ❞✉❛s ♠❛♥❡✐r❛s ❞❡ ❡①✐❜✐r♠♦s t❛✐s ❛①✐♦♠❛s✱ ✉♠❛ ❢♦r♠❛❧ ❡ ♦✉tr❛ q✉❡ ✉t✐❧✐③❛ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♠❛✐s ✉s✉❛❧✱ ♠❡♥♦s r✐❣♦r♦s❛✳ ❯t✐❧✐③❛r❡♠♦s ❛ s❡❣✉♥❞❛✱ ✈✐s❛♥❞♦ t♦r♥❛r ❛ ❧❡✐t✉r❛ ✉♠ ♣♦✉❝♦ ♠❡♥♦s té❝♥✐❝❛✿
• ❚♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ s✉❝❡ss♦r✱ q✉❡ t❛♠❜é♠ é ✉♠ ♥ú♠❡r♦
♥❛t✉r❛❧✳
• ◆ú♠❡r♦s ♥❛t✉r❛✐s ❞✐❢❡r❡♥t❡s ♣♦ss✉❡♠ s✉❝❡ss♦r❡s ❞✐❢❡r❡♥t❡s✳ ✭❖✉ ❛✐♥❞❛✿ ♥ú♠❡✲
r♦s q✉❡ tê♠ ♦ ♠❡s♠♦ s✉❝❡ss♦r sã♦ ✐❣✉❛✐s✳✮
• ❊①✐st❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥❡♥❤✉♠ ♦✉tr♦✳ ❊st❡
♥ú♠❡r♦ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ ✶ ❡ ❝❤❛♠❛❞♦ ❞❡ ✏♥ú♠❡r♦ ✉♠✑✳
✷✳✷✳ ❈❖◆❏❯◆❚❖❙ ❊◆❯▼❊❘➪❱❊■❙
• ❙❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♥té♠ ♦ ♥ú♠❡r♦ ✶ ❡✱ ❛❧é♠ ❞✐ss♦✱
❝♦♥té♠ ♦ s✉❝❡ss♦r ❞❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✱ ❡♥tã♦ ❡ss❡ ❝♦♥❥✉♥t♦ ❝♦✐♥❝✐❞❡ ❝♦♠ N✱ ✐st♦ é✱ ❝♦♥té♠ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
❖ ú❧t✐♠♦ ❛①✐♦♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ❡ é ♠✉✐t♦ út✐❧ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❡♥✈♦❧✈❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
❯♠ ♣r✐♥❝í♣✐♦ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ é ♦ ❝❤❛♠❛❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♠✱ q✉❡ ❛✜r♠❛ q✉❡ t♦❞♦ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ N ♣♦ss✉✐ ♠❡♥♦r ❡❧❡♠❡♥t♦✳
❆ ♣❛rt✐r ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s N✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦ ❝♦♥❥✉♥t♦
❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s Z ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s Q✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦
q✉♦❝✐❡♥t❡s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳
✷✳✷ ❈♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s
❉❡s❞❡ ♦s ♣r✐♠ór❞✐♦s✱ é ❝♦♠✉♠ ❡♥❝♦♥tr❛r ♦s ♠❛✐s ❞✐✈❡rs♦s t✐♣♦s ❞❡ ❛ss♦❝✐❛çõ❡s ❡♥tr❡ ♦❜❥❡t♦s ❡ q✉❛♥t✐❞❛❞❡s✳ ❖ s✐❣♥✐✜❝❛❞♦ ❞❡ss❛s s✐t✉❛çõ❡s é✱ s✐♠♣❧❡s♠❡♥t❡✱ ❝♦♥t❛r ♦✉✱ ❛✐♥❞❛✱ ❡♥✉♠❡r❛r✳
❉❡✜♥✐çã♦✿ ❯♠ ❝♦♥❥✉♥t♦ ❆✱ ✐♥✜♥✐t♦✱ é ❞✐t♦ ❡♥✉♠❡rá✈❡❧ s❡ é ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❝♦♠ ♦s ♥❛t✉r❛✐s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❆ é ❡♥✉♠❡✲ rá✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛ f :N→A✳
❊①❡♠♣❧♦s✿
• ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣❛r❡s P é ❡♥✉♠❡rá✈❡❧✿ ❢❛❝✐❧♠❡♥t❡
❡♥❝♦♥tr❛♠♦s ✉♠❛ ❜✐❥❡çã♦✳ ❇❛st❛ t♦♠❛r f :N→Pt❛❧ q✉❡ f(n) = 2n✳
• N×N é ❡♥✉♠❡rá✈❡❧✿ ❖ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❛❥✉❞❛ ❛ ❝♦♥str✉✐r ✉♠❛ ❜✐❥❡çã♦ ❞❡ N×N ❝♦♠ N✳
✷✳✷✳ ❈❖◆❏❯◆❚❖❙ ❊◆❯▼❊❘➪❱❊■❙
(1,1) //(1,2)
{
{
(1,3) //(1,4)
{
{
(1,5) //· · ·
|
|
(2,1)
(2,2)
;
;
(2,3)
{
{
(2,4)
;
;
(2,5)
{
{
· · ·
(3,1)
;
;
(3,2)
{
{
(3,3)
;
;
(3,4)
{
{
(3,5)
<
<
· · ·
|
|
(4,1)
(4,2)
;
;
(4,3)
{
{
(4,4)
;
;
(4,5)
{ { · · · ✳✳✳ ; ; ✳✳✳ ✳✳✳ ; ; ✳✳✳ ✳✳✳ ✳✳✳
• ❖ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s é ❡♥✉♠❡rá✈❡❧✿ P❛r❛ ✐st♦✱ ❞❡✜♥❛♠♦s ❛ ❢✉♥çã♦
f :N→Z ♣♦r
f(n) =
m s❡ n= 2m m = 1,2,3, . . .
−m s❡ n= 2m+ 1 m = 0,1,2,3, . . .
P♦r ❡①❡♠♣❧♦✱ f(1) = 0, f(2) = 1, f(3) =−1, f(4) = 2, f(5) =−2, . . .
• ❖ ❝♦♥❥✉♥t♦ ❞♦s r❛❝✐♦♥❛✐s é ❡♥✉♠❡rá✈❡❧ ❈♦♠♦ ❛❝✐♠❛✱ ♦ ❞✐❛❣r❛♠❛ q✉❡
s❡❣✉❡ ❛❥✉❞❛ ❛ ✈✐s✉❛❧✐③❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♣♦s✐t✐✈♦s Q+ é
❡♥✉♠❡rá✈❡❧✳
✷✳✷✳ ❈❖◆❏❯◆❚❖❙ ❊◆❯▼❊❘➪❱❊■❙
1 1 //
2 1
3 1 //
4 1
5
1 //· · ·
1 2 2 2 D D 3 2 4 2 D D 5 2 · · · 1 3 D D 2 3 3 3 D D 4 3 5 3 C C · · · 1 4 2 4 D D 3 4 4 4 D D 5 4 · · · ✳✳✳ C C ✳✳✳ ✳✳✳ C C ✳✳✳ ✳✳✳ ✳✳✳
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♥❡❣❛t✐✈♦s
Q− t❛♠❜é♠ é ❡♥✉♠❡rá✈❡❧✱ ♣♦✐s ❝♦♠♦
Q=Q+
∪Q−∪ {0}, ❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❣❛r❛♥t❡ q✉❡ Q é ❡♥✉♠❡rá✈❡❧✳
Pr♦♣♦s✐çã♦ ✶ ❆ ✉♥✐ã♦ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s é ❡♥✉♠❡rá✈❡❧✳
Pr♦✈❛ ❙❡❥❛♠ A={a1, a2, a3, . . .} ❡ b ={b1, b2, b3, . . .} ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s✳
❆ ❢✉♥çã♦ f :N→A∪B ❞❡✜♥✐❞❛ ♣♦r✿ f(n) =
an
2, s❡ n = 2k
bn+1
2 , s❡ n = 2k−1
k∈N
é s♦❜r❡❥❡t✐✈❛✳ ❆ ❛♣❧✐❝❛çã♦ ✐♥✈❡rs❛ ❞❛ ❞✐r❡✐t❛ ❞❡ f ♣❡r♠✐t❡ ✐❞❡♥t✐✜❝❛rA∪B ❝♦♠♦ ✉♠
s✉❜❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❞❡N❡ ♣♦rt❛♥t♦ A∪B é ❡♥✉♠❡rá✈❡❧✳
▼❛✐s ❣❡r❛❧♠❡♥t❡✱ é ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡ ❛ ✉♥✐ã♦ ✜♥✐t❛ ♦✉ ❡♥✉♠❡rá✈❡❧ ❞❡ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s é ❡♥✉♠❡rá✈❡❧✳
Pr♦♣♦s✐çã♦ ✷ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s R ♥ã♦ é ❡♥✉♠❡rá✈❡❧✿
Pr♦✈❛✿ ▼♦str❛r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ A = [0,1) ♥ã♦ é ❡♥✉♠❡rá✈❡❧✳ ❈♦♥s❡q✉❡♥✲
t❡♠❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s r❡❛✐s t❛♠❜é♠ ♥ã♦ s❡rá✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ ❆ t❡♠ ❛ s❡❣✉✐♥t❡ ❡①♣❛♥sã♦ ❞❡❝✐♠❛❧✿
