Độ đo hausdorff của các tập sierpinski

▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ✈➔ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐ ✹ ✶✳✶ ✣ë ✤♦✱ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❈→❝ t➟♣ ❙✐❡r♣✐♥s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❈❤÷ì♥❣ ✷✳ ▼ët sè t❤✉➟t t♦→♥ ✤→♥❤ ❣✐→ ữợ ữủ s r t rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ ✤➺♠ ❙✐❡r♣✐♥s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ t❤↔♠ ❙✐❡r♣✐♥s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✶ ▲❮■ ◆➶■ ✣❺❯ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✤ë ✤♦ q✉❛♥ trå♥❣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❤➻♥❤ ❤å❝ ❢r❛❝t❛❧✳ ❚✉② ♥❤✐➯♥✱ ✈✐➺❝ t➼♥❤ ❣✐→ trà ✤ë ✤♦ ♥➔② ❝õ❛ ❝→❝ t➟♣ ❢r❛❝t❛❧ ❧➔ r➜t ❦❤â✳ ❱➻ t❤➳✱ ❝❤♦ ✤➳♥ ♥❛② ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ t t ữợ ữủ tr ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ ❝→❝ t➟♣ ❧↕✐ ❝á♥ r➜t ➼t✱ t❤➟♠ ❝❤➼ ✈ỵ✐ ❝↔ ❝→❝ t➟♣ ❢r❛❝t❛❧ ❝ê ✤✐➸♥ ✈➔ tữớ tợ tớ t q t ✤÷đ❝ ✈➲ ✈✐➺❝ t➼♥❤ ❝❤➼♥❤ ①→❝ ❣✐→ trà ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝❤➾ ♠ỵ✐ ❞➔♥❤ ❝❤♦ t➟♣ ❈❛♥t♦r ✤➲✉✱ ✈➔ ♠ët số tt t ữợ t tr ữợ ✈➲ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝❤♦ ♠ët sè ❧ỵ♣ t➟♣ ❙✐❡r♣✐♥s❦✐✱✳✳✳ ❜ð✐ ❏✳ ▼❛r✐♦♥ ✭✶✾✽✼✮✱ ❩❤♦✉ ✭✶✾✾✼✮ ❲❛♥❣ ✭✶✾✾✾✮✱ ❈❤❡♥ ❉❛♥ ✈➔ ❨❛♥❣ ❳✐❛❧❛♦❧✐♥❤ ✭✷✵✵✹✮✱ ❙✳ ❙✉✱ ❲✳ ❙✉ ✈➔ ❩✳ ❩❤♦✉ ✭✷✵✵✽✮✱ ❆✳ ▼❛♥♥✐♥❣ ❛♥❞ ❑✳ ❙✐♠♦♥ ✭✷✵✵✾✮ ✈➔ ❣➛♥ ✤➙② ♥❤➜t ✈➔♦ ♥➠♠ ✷✵✶✵ ❜ð✐ ❲♦❧❢❣❛♥❣ ❑r❡✐t♠❡✐❡r✳ ❈→❝ t➟♣ ❧♦↕✐ ❙✐❡r♣✐♥s❦✐ ❧➔ ♥❤ú♥❣ t➟♣ ❢r❛❝t❛❧ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❤➻♥❤ ❤å❝ ❢r❛❝t❛❧✳ ❱✐➺❝ ✤÷❛ r❛ tt t ữợ t tr sr t õ ỵ ❝➛♥ t❤✐➳t✳ ❱➻ t❤➳✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ❧➔ ✧✣ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐✧✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ t❤ỉ♥❣ q✉❛ ❝→❝ t➔✐ ❧✐➺✉✱ ❝❤ó♥❣ tæ✐ t➻♠ ❤✐➸✉✱ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ✈➲ ❝→❝ t❤✉➟t t♦→♥ t➼♥❤ ❤❛② ✤→♥❤ ❣✐→ ❝➟♥ tr➯♥✱ ❝➟♥ ữợ sr t rs ỗ tớ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❝→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ♠➔ tr♦♥❣ t➔✐ ❧✐➺✉ ❝❤ù♥❣ ♠✐♥❤ ❝á♥ ✈➢♥ t➢t✳ ❚❤ỉ♥❣ q✉❛ ✤â ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈✐➺❝ ✈➟♥ ❞ư♥❣ ❝→❝ t❤✉➟t t♦→♥ ✤è✐ ✈ỵ✐ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐ ❝❤♦ ❝→❝ t➟♣ ❢r❛❝t❛❧ ❝ê ✤✐➸♥ t❤÷í♥❣ ❣➦♣ ❦❤→❝ ♥❤÷ ✤÷í♥❣ ❝♦♥❣ ❱♦♥ ❑♦❝❤✱ ❇ư✐ ❈❛♥t♦r✳✳✳ ợ õ ữủ tr ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ✈➔ t➟♣ ❙✐❡r♣✐♥s❦✐ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤ë ✤♦✱ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ✈➔ tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐✱ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ ♥➔②✳ ✷ ữỡ ởt số tt t ữợ ❧÷đ♥❣ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ t➟♣ ❙✐❡r♣✐♥s❦✐ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ t❤✉➟t t♦→♥ ✤➸ t➼♥❤ ❤❛② ÷ỵ❝ t➼♥❤ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐ ♥❤÷✿ t t sỷ {P n (S)} ữợ ❧÷đ♥❣ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ t❤↔♠ ❙✐❡r♣✐♥s❦✐❀ ❚❤✉➟t t♦→♥ ❞ü❛ trú t rs ữợ ữủ t t ữợ ữủ tr ữợ ❍❛✉s❞♦r❢❢ ❝õ❛ ✤➺♠ ❙✐❡r♣✐♥s❦✐❀ ❚❤✉➟t t♦→♥ t➼♥❤ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ t rs ợ ởt ữủ t t trữớ ữợ sỹ ữợ ổ ụ ỗ ❚❤❛♥❤✳ ❚→❝ ❣✐↔ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t tỵ✐ ❝ỉ✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ❝õ❛ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ t➟♥ t➻♥❤ ❞↕② ❞é✱ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t ✤➳♥ qỵ ổ rữớ ỗ ✤➻♥❤ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ị t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ♥➠♥❣ ❧ü❝✱ ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ❝→❝ t sõt t ữủ ỵ õ õ qỵ ổ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ❣✐↔ ✸ ❈❍×❒◆● ✶ ✣❐ ✣❖ ❍❆❯❙❉❖❘❋❋ ❱⑨ ❈⑩❈ ❚❾P ❙■❊❘P■◆❙❑■ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤ë ✤♦✱ ✤ë ✤♦ ❍❛✉s❞♦r❢❢✱ tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ ♥➔②✳ ✶✳✶✳ ✣❐ ✣❖✱ ✣❐ ✣❖ ❍❆❯❙❉❖❘❋❋ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ ❝➛♥ ❞ị♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ X ❧➔ ♠ët t ủ tũ ỵ C ởt số ❝→❝ t➟♣ ❤đ♣ ❝♦♥ ❝õ❛ X ✳ ❍➔♠ t➟♣ µ : C −→ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤ë ✤♦ tr➯♥ C ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✐✮ µ(A) ≥ ∀A ∈ C ❀ ✐✐✮ µ(∅) = 0❀ ✐✐✐✮ µ ❧➔ σ ✲ ❝ë♥❣ t➼♥❤✱ tù❝ ❧➔ ♥➳✉ ∞ Ai ∈ C, i = 1, 2, , Ai ∩ Aj = ∅, i = j, Ai ∈ C t❤➻ i=1 ∞ µ( ∞ Ai ) = i=1 µ(Ai ) i=1 ✣ë ✤♦ µ tr➯♥ σ ✲ ✤↕✐ sè L ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤ë ✤♦ ✤õ ♥➳✉ A ⊂ B, B ∈ L, µ(B) = t❤➻ A ∈ L, µ(A) = 0✳ ữủ ởt tr C ♥➳✉ t❤ä❛ ♠➣♥ ✐✮✱ ✐✐✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ✐✐✐✮ ữủ t ợ ữợ t tự Ai C, i = 1, 2, , ∞ Ai ∈ C t❤➻ i=1 ∞ µ( ∞ Ai ) ≤ i=1 µ(Ai ) i=1 ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ U ❧➔ ♠ët t➟♣ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ (Rn , d)✱ ✤÷í♥❣ ❦➼♥❤ U ỵ |U | ữỡ ✤à♥❤ ❜ð✐ |U | = sup{d(x, y) : x, y U } ỵ d(x, y) ữủ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ t❤ỉ♥❣ t❤÷í♥❣ ❣✐ú❛ ❤❛✐ ♣❤➛♥ tû x ✈➔ y tr♦♥❣ Rn ✳ ❱ỵ✐ ♠é✐ δ > ✈➔ Ui ⊂ Rn , i = 1, 2, ❤å {Ui }∞ i=1 ✤÷đ❝ ❣å✐ ❧➔ δ ✲ ♣❤õ U ♥➳✉ < |Ui | ≤ δ, i = 1, 2, ✈➔ ∞ Ui ⊃ U ✳ i=1 ●✐↔ sû F ❧➔ ♠ët t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ tr♦♥❣ Rn ✈➔ s ❧➔ ♠ët sè t❤ü❝ ❦❤ỉ♥❣ ➙♠✳ ❱ỵ✐ ộ > t ỵ Hs (F ) |Ui |s : {Ui } ❧➔ δ − ♣❤õ F = inf ✭✶✳✶✮ i=1 ✶✳✶✳✸✳ ◆❤➟♥ ①➨t✳ ●✐↔ sû r➡♥❣ F ⊂ Rn✱ s > ✈➔ δ > 0✱ ❦❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦➳t ❧✉➟♥ s❛✉ s❛✉✳ ✐✮ Hδs (F ) > 0✳ ✐✐✮ ◆➳✉ < δ < δ t❤➻ Hδs (F ) ≥ Hδs (F ) ữ ợ ộ s trữợ Hs (F ) t õ tỗ t lim+ Hs (F ) ợ F ⊂ Rn ❞ị ❣✐ỵ✐ ❤↕♥ ✤â ❝â t❤➸ ❧➔ ❤❛② +∞✳ ✣➦t δ→0 Hs (F ) = lim+ Hδs (F ) δ→0 ✭✶✳✷✮ ✶✳✶✳✹✳ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ❈❤♦ C ❧➔ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ Rn✱ ✈ỵ✐ ♠é✐ s > ❤➔♠ t➟♣ Hs : C → R ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✷✮ ✈ỵ✐ ♠å✐ F ∈ C ❧➔ ♠ët ✤ë ✤♦ ♥❣♦➔✐ tr➯♥ C ✳ ✶✳✶✳✺✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ✣ë ✤♦ s✐♥❤ ❜ð✐ ✤ë ✤♦ ♥❣♦➔✐ Hs ✤÷đ❝ ❣å✐ ❧➔ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ tr➯♥ δ ✲ ✤↕✐ sè L ❝→❝ t➟♣ ❝♦♥ Hs ✲ ✤♦ ✤÷đ❝ ❝õ❛ Rn ✈➔ ỵ Hs F Rn t❤ä❛ ♠➣♥ < Hs < +∞ ✤÷đ❝ ❣å✐ ❧➔ s t➟♣✳ ✶✳✶✳✻✳ ◆❤➟♥ ①➨t✳ ✐✮ ❚r♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✤ë ✤♦ ❍❛✉s❞♦r❢❢✱ ❝â t❤➸ t❤❛② ♣❤õ ❜➜t ❦ý ❜➡♥❣ ♣❤õ ♠ð ✭♣❤õ ✤â♥❣✮✳ ✐✐✮ ◆➳✉ F ❧➔ t➟♣ ❝♦♠♣❛❝t t❤➻ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝â t❤➸ t❤❛② ♣❤õ ❜➜t ❦ý ❜➡♥❣ ♣❤õ ♠ð ❤ú✉ ❤↕♥✳ ✶✳✶✳✼✳ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ✐✮ ◆➳✉ F ⊂ Rn ✈➔ λ > t❤➻ Hs(λF ) = λsHs(F ) ✈ỵ✐ ♠é✐ s ≥ ✈➔ λF = {λx : x ∈ F } ✺ ✐✐✮ ◆➳✉ F ⊂ Rn ✈➔ f : F → Rn ❧➔ ♠ët →♥❤ ①↕ ❍♦❧❞❡r✱ t❤ä❛ ♠➣♥ |f (x) − f (y)| ≤ c|x − y|α , ∀x, y ∈ F ✈ỵ✐ ❝→❝ ❤➡♥❣ sè c > > trữợ õ ợ s ≥ t❛ ❝â Hs/α (f (F )) ≤ cs/α Hs (F ) ✶✳✶✳✽✳ ❍➺ q✉↔ ✭❬✸❪✮✳ ✐✮ ◆➳✉ α = t❤➻ f ❧➔ →♥❤ ①↕ ▲✐♣s❝❤✐t③✳ ❑❤✐ ✤â✱ Hs (f (F )) ≤ cs Hs (F ) ✐✐✮ ◆➳✉ f : F → Rn ❧➔ ♣❤➨♣ ✤➥♥❣ ❝ü tø F ❧➯♥ f (F )✱ ♥❣❤➽❛ ❧➔ |f (x) − f (y)| = |x − y|, ∀x, y ∈ F t❤➻ Hs (f (F )) = Hs (F ) ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❚ø ▼➺♥❤ ✤➲ ✶✳✶✳✼✳✐✐✮ t❤❛② α = t❛ ✤÷đ❝ Hs (f (F )) ≤ cs Hs (F ) ✐✐✮ ❱➻ |f (x) − f (y)| ≤ |x − y| ♥➯♥ t❤❡♦ ✐✮ t❛ ❝â Hs (f (F )) ≤ Hs (F ) ✭✶✮ ▼➦t ❦❤→❝✱ ✈➻ f ỹ tỗ t f ❝â t❤➸ ✈✐➳t F t❤❛② ❜ð✐ f (F ) t❛ ✤÷đ❝ Hs (f −1 (f (F ))) ≤ Hs (f (F )) ✭✷✮ ❚ø ✭✶✮ ✈➔ ✭✷✮✱ s✉② r❛ Hs (f (F )) ≤ Hs (F )✳ ✶✳✶✳✾✳ ❇ê ✤➲ ✭❬✸❪✮✳ ❱ỵ✐ δ > 0, t > s ≥ ✈➔ F ⊂ Rn t❛ ❧✉æ♥ ❝â Hδt (F ) ≤ δ t−s Hδt (F ) ✻ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {Ui } ❧➔ δ− ♣❤õ ❝õ❛ F ✳ ❱ỵ✐ t > s ≥ t❛ ❝â s✉② r❛ |Ui | δ ❞➝♥ ✤➳♥ |Ui | ≤ δ t t−s t |Ui | ❱➟② t❛ ❝â s ▲➜② ✐♥❢✐♠✉♠ ❤❛✐ ✈➳ t❛ ❝â < ∞ |Ui | δ t |Ui | δ s ≥ t❛ ❝â ✲ ◆➳✉ Hs (F ) < +∞ t❤➻ Ht (F ) = ✲ ◆➳✉ Hs (F ) > t❤➻ Ht (F ) = +∞ ✶✳✶✳✶✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ D = φ✱ D ⊂ Rn✱ ✈➔ →♥❤ ①↕ f : D → D ✐✮ f ✤÷đ❝ ❣å✐ ❧➔ →♥❤ tr D tỗ t c (0; 1) ✤➸ |f (x) − f (y)| = c|x − y|, ∀x, y ∈ D, c ✤÷đ❝ ❣å✐ ❧➔ t✛ sè ❝♦ ❝õ❛ →♥❤ ①↕ f ✳ ✐✐✮ f ✤÷đ❝ ỗ tr D tỗ t↕✐ c ∈ (0; 1) ✤➸ |f (x) − f (y)| = c|x − y|, ∀x, y ∈ D, c ữủ t số ỗ f ✳ ✶✳✶✳✶✷✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ▼ët ❤å ❤ú✉ ❤↕♥ ỗ m {f1, f2, , fm} tr➯♥ D ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ❤➔♠ ❧➦♣ ✭✈✐➳t t➢t ❧➔ ■❋❙ ✲ ■t❡r❛t❡❞ ❋✉♥❝t✐♦♥ ❙②st❡♠✮ tr➯♥ D✳ ✶✳✶✳✶✸✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ A ❧➔ ♠ët t➟♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ✐✮ ❱ỵ✐ ♠é✐ ✤✐➸♠ x ∈ X t❛ ✤➦t d(x, A) = inf{d(x, y) : y ∈ A} ✈➔ t❛ ❣å✐ d(x, A) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø x ✤➳♥ t➟♣ A✳ ✐✐✮ ❈❤♦ δ > 0✱ t➟♣ Aδ = {x ∈ X : d(x, A) } ỗ ỳ t A ởt ❦❤æ♥❣ q✉→ δ ✱ ❣å✐ ❧➔ δ ✲❜❛♦ ❝õ❛ A✳ ✼ ✶✳✶✳✶✹✳ ◆❤➟♥ ①➨t✳ ◆➳✉ A, B ❧➔ ❤❛✐ t➟♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ (X, d) t❤➻ B ⊂ Aδ ❝â ♥❣❤➽❛ ❧➔ ♠å✐ ✤✐➸♠ ❝õ❛ B ✤➲✉ ❝→❝❤ A ♠ët ổ q ỵ D ❧➔ t➟♣ ❝♦♠♣❛❝t ❦❤→❝ ré♥❣ tr♦♥❣ Rn✳ ❑➼ ❤✐➺✉ K ❧➔ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t✱ ❦❤→❝ ré♥❣ ❝õ❛ ❉✳ ❑❤✐ ✤â✱ ❤➔♠ dH : K × K → R (A, B) → dH (A, B) = inf{δ ≥ : A ⊆ Bδ , B ⊆ Aδ } t❤ä❛ ♠➣♥ ✐✮ dH (A, B) = max sup d(x, B); sup d(y, A) ✳ x∈A y∈B ✐✐✮ dH ❧➔ ♠ët ♠➯tr✐❝ tr➯♥ K✳ ❍ì♥ ♥ú❛✱ ❦❤ỉ♥❣ ❣✐❛♥ (K, dH ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✳ ✶✳✶✳✶✻✳ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ❈❤♦ ♠ →♥❤ ①↕ ❝♦ {Si}mi=1 tr➯♥ ❉✳ ❚❛ ①→❝ ✤à♥❤ →♥❤ ①↕ S : K → K ❜ð✐ m E → S(E) = ✭✶✳✷✮ Si (E) i=1 t❤➻ dH (S(A), S(B)) ≤ cmax dH (A, B) ✈ỵ✐ cmax = max {ci }✱ ci ❧➔ t✛ sè ❝♦ ❝õ❛ 1≤i≤m →♥❤ ①↕ Si , i = 1, , m ỵ ❤➔♠ ❧➦♣ {Si}mi=1 ✈➔ →♥❤ ①↕ S ✤÷đ❝m ①→❝ ✤à♥❤ õ tỗ t t ởt t F ∈ K s❛♦ ❝❤♦ F = Si (F )✳ i=1 ∞ ❍ì♥ ♥ú❛✱ ♥➳✉ ❝â t➟♣ E ∈ K s❛♦ ❝❤♦ Si (E) ⊆ E, i ≤ ≤ m t❤➻ F = S k (E) ✈ỵ✐ S k ❧➔ sü ❧➦♣ ❧↕✐ k ❧➛♥ ❝õ❛ →♥❤ ①↕ S, k = 1, 2, 3, k=1 ✶✳✶✳✶✽✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ✐✮ ❈❤♦ ❤➺ ❤➔♠ ❧➦♣ {Si}mi=1 tr➯♥ D✳ ❚➟♣ F ữủ ữ tr ỵ ữủ ❣å✐ ❧➔ t➟♣ ❜➜t ❜✐➳♥ ❤❛② t➟♣ ❤ót ❝õ❛ ❤➺ ❤➔♠ ❧➦♣ {Si }m i=1 ✳ ✐✐✮ ◆➳✉ ❝→❝ Si (1 i m) ỗ ❞↕♥❣ t❤➻ t➟♣ ❜➜t ❜✐➳♥ F ✤÷đ❝ ❣å✐ ❧➔ t➟♣ tỹ ỗ õ r m →♥❤ ①↕ ❝♦ {f1, f2, , fm} t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð ✭❖❙❈ ✲ ❖♣❡♥ t t tỗ t t V ré♥❣ ✤➸  V ⊃ m f (V ) i i=1  fi (V ) ∩ fj (V ) = ∅, ∀i = j ✭✶✳✸✮ ❚ø ◆❤➟♥ ①➨t ✶✳✶✳✶✵✱ t❛ ❝â ♠➺♥❤ ✤➲ s❛✉✳ ✶✳✶✳✷✵✳ ▼➺♥❤ ✤➲ ✭❬✸❪✮✳ ❈❤♦ F Rn õ tỗ t t ởt ❣✐→ trà sF ∈ [0, +∞] s❛♦ ❝❤♦ ✐✮ Hs (F ) = ✈ỵ✐ ♠å✐ s > sF ✳ ✐✐✮ Hs (F ) = ∞ ✈ỵ✐ ♠å✐ s < sF ✳ ✶✳✶✳✷✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ F ⊂ Rn✱ ❝❤✐➲✉ ❍❛✉s❞♦r❢❢ ❝õ❛ F ❧➔ ❣✐→ trà sF ∈ [0; +] õ tr ỵ ❞✐♠H F ✳ ◆❤÷ ✈➟②✱ dimH F = inf{s ≥ : Hs (F ) = 0} = sup{s ≥ : Hs (F ) = } ỵ ✭❬✸❪✮✳ ❈❤♦ {S1, S2, , Sm} ❧➔ m →♥❤ ①↕ ỗ ợ t số ci (1 i m) t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð✳ ❑❤✐ ✤â✱ ♥➳✉ F t tỹ ỗ ♥❣❤➽❛ ❧➔ F = ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ m i=1 m Si (F ) t❤➻ dimH F = s ✈ỵ✐ s ❧➔ i=1 csi = ❍ì♥ ♥ú❛✱ ✈ỵ✐ ❣✐→ ♥➔② ❝õ❛ s t❛ ❝â < H (F ) < +∞ s ✶✳✶✳✷✸✳ ▼➺♥❤ ✤➲ ✭❬✽❪✮✳ ●✐↔ sỷ E ởt t tỹ ỗ s ❤➺ ❤➔♠ ❧➦♣ t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➟♣ ♠ð✳ ❑❤✐ õ ợ t ý t ữủ U, t õ Hs (E ∩ U ) ≤ |U |s ✈ỵ✐ s = dimH (E) ✾ ✶✳✷✳ ❈⑩❈ ❚❾P ❙■❊❘P■◆❙❑■ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥â✳ ✶✳✷✳✶✳ ✣➺♠ ❙✐❡r♣✐♥s❦✐ ✭❙✐❡r♣✐♥s❦✐ ❣❛s❦❡t✮ ✶✳✷✳✶✳✶✳ ❳➙② ❞ü♥❣ ✤➺♠ ❙✐❡r♣✐♥s❦✐ ✣➺♠ ❙✐❡r♣✐♥s❦✐ S ởt tr ỳ t tỹ ỗ ◆â ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➭♣ ✈➔ ✤÷đ❝ ❧➜② ❧➔♠ ✈➼ ❞ư ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➟♣ ❋r❛❝t❛❧✳ ◆â ✤÷đ❝ ①➙② ❞ü♥❣ ♥❤÷ s❛✉✳ ❈❤♦ ♠ët t❛♠ ❣✐→❝ ✤➲✉ ABC ✈ỵ✐ ❝❤✐➲✉ ❞➔✐ ❝õ❛ ❝↕♥❤ ❜➡♥❣ ✶✱ ❣å✐ ❧➔ S0 ◆è✐ ❝→❝ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❝→❝ ❝↕♥❤ ✈➔ ❜ä t ữủ ợ t ❣✐→❝ ❦❤→❝✳ ●å✐ t➟♣ ❝á♥ ❧↕✐ ❧➔ S1 ❱ỵ✐ ♠é✐ ♠ët tr♦♥❣ ❜❛ ❤➻♥❤ t❛♠ ❣✐→❝ ❝á♥ ❧↕✐ ❝õ❛ S1 ♥è✐ ❝→❝ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❝→❝ ❝↕♥❤ ✈➔ ❜ä t ữủ ợ t❛♠ ❣✐→❝ ❝á♥ ❧↕✐✳ ●å✐ t➟♣ ❝á♥ ❧↕✐ S2 ▲➦♣ ✤✐ ❧➦♣ ❧↕✐ q✉→ tr➻♥❤ ♥➔②✱ t❛ ✤÷đ❝ S0 ⊃ S1 ⊃ S2 ⊃ ⊃ Sn ⊃ ❚➟♣ ❤đ♣ ❦❤→❝ ré♥❣ S = Sn ✤÷đ❝ ❣å✐ ❧➔ rs rs n0 st é ữợ tự n, Sn ỗ õ 3n t õ ❤✐➺✉ ❧➔ ∆n1 , ∆n2 , , ∆n3n ▼é✐ ∆ni ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ t❛♠ ❣✐→❝ ❜➟❝ n 2n ỵ ỵ sû F ❧➔ ✤➺♠ ❙✐❡r♣✐♥s❦✐✱ ❦❤✐ ✤â dimH F = log log ✶✵ ●✐↔ sû a(3) n = 2≤k≤2.3n−1     | ∆i ∈Sn i=1,2, ,k         ∆i |s k  i=1 : ( ∆i ) ⊂ (∆13 ∪ ∆22 )  k( 13 )n  i=1   k (3) (3) ❈→❝ {an } ❣✐↔♠ ✈➔ ❣✐↔ sû r➡♥❣ lim an = α3 ✱ t❛ ❝â n→∞ √ − 163 s( 21 )n a(3) n e α3 ≥ k ❚r÷í♥❣ ❤đ♣ ✸✳ ◆➳✉ √ − 163 s( 12 )n ≥ an e ∆i ❝❤➾ ❣✐❛♦ ✈ỵ✐ ♠ët t❛♠ ❣✐→❝ ✤➲✉ ❝ì sð i=1 S1 tỗ t ởt số ữỡ t0 ✈➔ i01 , i02 , , i0n ∈ {1, 2, 3} s❛♦ ❝❤♦ k i=1 ∆i ⊂ Ei01 i02 i0n = fi01 ◦ fi02 ◦ ◦ fi0n (S) ✈➔ k ∆i ❣✐❛♦ ✈ỵ✐ ➼t ♥❤➜t ❤❛✐ tr♦♥❣ i=1 ◦ ◦ fi−1 ◦ fi−1 sè Ei01 i02 i0n , Ei01 i02 i0t , Ei01 i02 i0t ❃ ❉♦ ✤â fi−1 0 ( 0 t0 ➼t ♥❤➜t ❤❛✐ tr♦♥❣ sè f1 (S), f2 (S), f3 (S) ỵ r k | |fi1 ◦ ◦ fi−1 ◦ fi−1 0 ( ∆i |s i=1 k = µ( ∆i ) i=1 ✈➔ fi−1 t0 ◦ ◦ fi−1 ◦ t0 ◦ ◦ fi−1 ◦ µ(fi−1 0 t0 k −1 fi0 ( ∆i ) i=1 k k ∆i ) ❣✐❛♦ ✈ỵ✐ i=1 ∆i )|s i=1 k fi−1 ∆i )) ( i=1 t❤ä❛ ♠➣♥ ❚r÷í♥❣ ❤đ♣ ✶ ❤♦➦❝ ❚r÷í♥❣ ❤đ♣ ✷ (1) ✭t÷ì♥❣ tü ♥❤÷ an ❝õ❛ ❚r÷í♥❣ ❤đ♣ ✶✮✳ ●✐↔ sû a(4) n = 1≤k≤3n−1 ∆i ∈Sn i=1,2, ,k     |         ∆i |s k  i=1 0 : ( ∆i ) ⊂ ∆i01 , i1 ∈ {1, 2,  k( 13 )n  i=1   k (4) (4) ❈→❝ {an } ❣✐↔♠ ✈➔ ❣✐↔ sû r➡♥❣ lim an = α4 ✱ t❛ ❝â n→∞ − α4 ≥ a(4) n e √ n 16 n −8s( 2) s( ) e ✷✼ ≥ an e− √ 16 n s( ) ❚÷ ❝→❝ ❚r÷í♥❣ ❤đ♣ ✶ ✈➔ ✸ t❛ ❝â Hs (S) = lim an n→∞ (1) (2) (3) (4) = lim min{an , an , an , an } n→∞ = min{α1 , α2 , α3 , α4 } √ √ √ −8s( 21 )n − 163 s( 21 )n − 163 s( 12 )n − 163 s( 12 )n ≥ an e , an e , an e , an e √ = an e ✳ ≤ Hs (S) ≤ an ợ n ỵ ữủ ự − 163 s( 21 )n √ − 163 s( 21 )n ❱➻ ✈➟② an e ✷✳✶✳✷✳✺ ❍❛✐ ❞ü ✤♦→♥ ✈➲ ✤➺♠ ❙✐❡r♣✐♥s❦✐ ✭❬✽❪✮✳ s ❚ø ❦➳t q✉↔ ❝õ❛ ✣à♥❤ ❧➼ ✷✳✷✳✶✳✶ t❛ ❦✐➸♠ tr❛ ✤÷đ❝ a1 = 1, a2 = 36 ≈ 0.9508✳ ❝á♥ ❦❤✐ n > ✈➝♥ ❝❤÷❛ ①→❝ ✤à♥❤ ✤÷đ❝ ❝ư t❤➸ an ✳ ◆➠♠ ✷✵✵✼✱ ❇❛♦❣✉❛ ❏✐❛ ❞ü ✤â♥ r➡♥❣ ❉ü ✤♦→♥ ✶✳ ✣è✐ ✈ỵ✐ ✤➺♠ ❙✐❡r♣✐♥s❦✐✱ s 7s a3 = 24 ≈ 0.91047736 ✭①❡♠ ❤➻♥❤ ✻✮✱ a4 = 13 66 ≈ 0.88319434✱ 25s 49s a5 = 192 ≈ 0.85592100617✱ a6 = 570 ≈ 0.83769501528✱ 97s 193s a7 = 1698 ≈ 0.8300332938336✱ a8 = 5082 ≈ 0.825227465852 ❉ü ✤♦→♥ ✷✳ ✣ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ ✤➺♠ ❙✐❡r♣✐♥s❦✐ t❤ä❛ ♠➣♥ √ 0.779355 ≈ a8 e − 163 s( 12 )8 ❤✐♥❤ ✹✱✺✱✻✱✼ ✷✽ ≤ Hs (S) ≤ a8 ✷✳✷✳ ✣❐ ✣❖ ❍❆❯❙❉❖❘❋❋ ❈Õ❆ ❚❍❷▼ ❙■❊❘P■◆❙❑■ ✷✳✷✳✶ ❈→❝ ❦➳t q✉↔ ❜ê trñ ✭❬✹❪✮✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❜ê trđ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ ✈➲ ✈✐➺❝ t➼♥❤ ✤ë ✤♦ ❍❛✉s❞♦r❢❢ ❝õ❛ t❤↔♠ ❙✐❡r♣✐♥s❦✐✳ ✷✳✷✳✶✳✶ ❇ê ✤➲ ✭❬✹❪✮✳ ợ ữợ HFs (E) ữủ ỹ ữ ð ▼ö❝ s ✶✳✷✳✷✱ t❛ ❝â HFs (E) = 2 = |E0 |s ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ G = {Ui } ởt ữủ tũ ỵ ❝õ❛ E ❜ð✐ ❝→❝ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ HFs (E) t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ ♠é✐ ♣❤➛♥ tû tr♦♥❣ G ❦❤æ♥❣ t❤➸ ❝❤ù❛ ❤♦➔♥ t♦➔♥ ♠ët ♣❤➛♥ tû ❦❤→❝✳ ▲➜② k = min{j : Ui ∈ F, Ui ∈ G} ◆➳✉ ♠å✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ó k ✤➯✉ ♥➡♠ tr♦♥❣ G t❤➻ √ s |Ui |s = 4k ( 2λk )s = 2 , Ui ∈G ✈ỵ✐ ♠å✐ s ❧➔ ❝❤✐➲✉ ❍❛✉s❞♦r❢❢ ❝õ❛ E ✳ ◆➳✉ ❝â lk ✈ỵ✐ (0 < lk < 4k ) ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù k t❤✉ë❝ ✈➔♦ G t❤➻ s➩ ❝â (4k − lk ) ❤➻♥❤ ✈✉æ♥❣ ❝ì sð ❦❤ỉ♥❣ t❤✉ë❝ G ✳ ❇➙② ❣✐í✱ t❛ ①➨t ổ ỡ s ữợ tự k ổ t❤✉ë❝ (k + 1) ♠➔ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ 4k − lk ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù k ✳ ◆➳✉ ♥❤ú♥❣ ❤➻♥❤ ✈✉æ♥❣ ♥➔② t❤✉ë❝ G ✱ t❤➻ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ √ √ s |Ui |s = lk ( 2λk )s + 4(4k − lk )( 2λ(k+1) )s = 2 Ui G ữỡ tỹ tỗ t↕✐ ♠ët sè ♥❣✉②➯♥ t > s❛♦ ❝❤♦ t = max{j : Ui ∈ s F, Ui ∈ G}✱ t❛ ❝â |Ui |s = 2 ✳ Ui ∈G t ữ tr ổ tỗ t t s tỗ t↕✐ ❝→❝ sè ♥❣✉②➯♥ lm ✈ỵ✐ m = k, k + 1, ✈➔ ≤ lm < 4m s❛♦ ❝❤♦ ∞ s |Ui | = Ui ∈G m=k ✷✾ √ lm ( 2λm )s ❇➙② ❣✐í✱ t❛ ❦➼ ❤✐➺✉ np ❧➔ sè ❝→❝ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù p ♠➔ ❦❤ỉ♥❣ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❜➜t ❦➻ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù q ∈ G ✈ỵ✐ q < p✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ❝â ✈ỵ✐ ❜➜t ❦➻ sè ♥❣✉②➯♥ p > k t❤➻ p−1 √ √ s lm ( 2λm )s + np ( 2λp )s = s m=k √ ❚❛ ❝â ❦❤➥♥❣ ✤à♥❤ np ( 2λp )s −→ ❦❤✐ p −→ ∞✳ ❚❤➟t ✈➟②✱ ổ ữ s tỗ t số > ✈➔ ♠ët t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ N ỗ số ữỡ s np ( 2λp )s > γ ✈ỵ✐ p ∈ N✳ ❱ỵ✐ ♠é✐ p ∈ N✱ ❧➜② Wp ❧➔ ❤ñ♣ ❝õ❛ np ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù p ♠➔ ❦❤ỉ♥❣ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❜➜t ❦➻ ❤➻♥❤ ✈✉æ♥❣ ❝ð sð t❤ù q tr♦♥❣ G ✈ỵ✐ q < p✳ ❍✐➵♥ ♥❤✐➯♥ r➡♥❣ Wp ❧➔ t➟♣ ❝♦♠♣❛❝t✱ ❦❤→❝ ré♥❣ ✈➔ Wn ⊃ Wm ✈ỵ✐ m > n✳ ◆❤÷ ✈➟②✱ Wp = ∅ E⊂ p∈N ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Wp t❛ ❝â Wp ) ∪ ( ( Ui ) = ∅ Ui ∈G p∈C ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➞♥ ✈ỵ✐ t❤ü❝ t➳ G ❧➔ ♣❤õ ❝õ❛ E ✳ ❉♦ ✤â✱ t❛ ❝â ❦❤➥♥❣ ✤à♥❤ ❧➔ ✈ỵ✐ ♠é✐ δ > 0✱ t❤➻ s inf |Ui | = 2 Ui G ợ ữủ tr tt δ ♣❤õ ❝õ❛ E ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✤ë ữợ ữủ ự ✭❬✸❪✮✳ ●✐↔ sû λ ≤ 14 ✈➔ U ❧➔ ♠ët t➟♣ ❝♦♥ ♠ð ❝õ❛ R2 ✈ỵ✐ √ √ 2λk ✤è✐ ✈ỵ✐ sè ♥❣✉②➯♥ k > 0✳ ●✐↔ sû FU ❧➔ ♠ët ❤å ❝→❝ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð fδ , (fδ ∈ F ) ♠➔ ❤♦➔♥ t♦➔♥ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ U ✈➔ ❦❤ỉ♥❣ 2λk+1 ≤ |U | ≤ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❝→✐ ❦❤→❝✳ ❑❤✐ ✤â✱ t❛ ❝â |U |s ≥ |fδ |s fδ ∈FU ✸✵ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ λ < 14 , λk−1 − 2λk ≥ √ 2λk > |U | ♥➯♥ U ❝â t❤➸ ❣✐❛♦ ✈ỵ✐ ❝❤➾ ♠ët ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù k ❦❤ỉ♥❣ ❤♦➔♥ t♦➔♥ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ U ✳ ❈❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù k ❣✐❛♦ ✈ỵ✐ U ❧➔ A = [0, λk ]x[0, λk ]✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ t❤❡♦ ❜è♥ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙②✳ ❚r÷í♥❣ ❤đ♣ ✶✳ U ❣✐❛♦ ✈ỵ✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + 1) tr♦♥❣ A✳ ❚÷ì♥❣ tü ♥❤÷ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✷✳✶✳✶✱ t❛ ❝â √ |fδ |s ≤ |fk+1 |s = 2λk+1 ≤ |U |s fδ ∈FU ❚r÷í♥❣ ❤đ♣ ✷✳ U ❣✐❛♦ ✈ỵ✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + 1) tr♦♥❣ A✳ ●✐↔ sû In ✭t÷ì♥❣ ù♥❣ In )(n > k) ❧➔ ❝→❝ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù n ♣❤➼❛ tr➯♥ ❜➯♥ ♣❤↔✐ ✭t÷ì♥❣ ù♥❣ ❜➯♥ tr→✐✮ tr♦♥❣ A ✈➔ Jn ✭t÷ì♥❣ ù♥❣ Jn (n > k)✮ ổ ỡ s tự n ữợ tr→✐ ✭t÷ì♥❣ ù♥❣ ❜➯♥ ♣❤↔✐✮ tr♦♥❣ A✳ ❑❤↔ ♥➠♥❣ ✶✳ ●✐↔ sû r➡♥❣ U ∩ Jk+1 = ∅ ✈➔ U ∩ Ik+1 = ∅ ✭♥❤÷ ❜➯♥ tr→✐ ❝õ❛ ❍➻♥❤ ✽✮✳ ❉➵ ❞➔♥❣ ❝❤➾ r❛ r➡♥❣ √ |U | > |fk | − 2|fk+1 | = 2λk (1 − 2λ) ✈➔ t÷ì♥❣ tü ♥❤÷ ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✷✳✷✳✶✳✶✱ t❛ ❝â √ |fδ |s ≤ 2|fk+1 |s = ( λ)s f ∈F δ U ●✐↔ sû g(λ) = (1 − 2λ)s ✳ ❱➻ g(λ) ❣✐↔♠ tr➯♥ (0 41 ] ✈➔ g( 41 ) = 12 ✱ ♥➯♥ t❛ ❝â✱ ❦❤✐ λ ≤ 41 t❤➻ (1 − 2λ)s ≥ 12 ✈➔ √ √ k ( 2λk )s (1 − 2λ)s ≥ ( λ )s ❉♦ ✤â✱ |U |s ≥ fδ ∈ FU |fδ |s ❑❤↔ ♥➠♥❣ ✷✳ ●✐↔ sû r➡♥❣ U ∩ Jk + = ∅ ✈➔ U ∩ Jk + = ∅ ✭♥❤÷ ð ❜➯♥ ♣❤↔✐ ❝õ❛ ❍➻♥❤ ✽✮✳ ✸✶ ❇➙② ❣✐í t❛ ❣✐↔ sû a = min{x : (x, y) ∈ U ∩ Jk + 1}, (0 ≤ a ≤ λ1+k ) b = Max{x : (x, y) ∈ U ∩ Jk + 1}, (λk − λk+1 ≤ b ≤ λk ), a = λ1+k − a ✈➔ b = λk − b✳ ◆➳✉ < a ≤ λ1+k − λ2+k ✈➔ < b ≤ λ1+k − λ2+k ✱ t❤➻ t❛ ❝â u > λk − 2λ1+k ✈➔ √ |fδ |s ≤ 4|fk+2 |s = 4( 2λ2+k )s fδ ∈FU ❤➻♥❤ ✽✳ ◆❤÷ ✈➟②✱ t÷ì♥❣ tü ❑❤↔ ♥➠♥❣ ✶✱ t❤➻ ❦❤✐ λ ≤ 41 ✱ t❛ ❝â √ (λk − 2λ1+k )s > 4( 2λ2+k )s ✈➔ ❞♦ ✤â |U |s ≥ |fδ |s fδ ∈FU ◆➳✉ a > λ1+k − λ2+k ✈➔ < b ≤ λ1+k − λ2+k ✱ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ |U | ≥ λk − 2λ1+k + a + b > λk − 2λ1+k + (λ1+k − 2λ2+k ) ✸✷ ✈➔ √ |fδ |s ≤ 6|fk+2 |s = 6( 2λ2+k )s fδ ∈FU ❱➻ ✈➟②✱ ♥➳✉ λ ≤ t❛ ❝â √ λk − 2λ1+k + λ1+k − 2λ2+k ≥ 6( 2λk+2 )s ✈➔ ✈➻ ✈➟② |U |s ≥ |fδ |s fδ ∈FU ◆➳✉ a > λ1+k − λ2+k ✈➔ b > λ1+k − λ2+k t❛ ❝â |U | ≥ λk − 2λ1+k + a + b > λk − 2λ2+k ✈➔ √ 2+k s |fδ | ≤ 8|fk+2 | = 8( 2λ ) s s fδ ∈FU ❱➻ ✈➟②✱ ❦❤✐ λ ≤ 41 ✱ t❤➻ √ λk − 2λ2+k ≥ 8( 2λ2+k )s ✈➔ ❞♦ ✈➟② |U |s ≥ |fδ |s fδ ∈FU ❚r÷í♥❣ ❤đ♣ ✸✳ U ❣✐❛♦ ✈ỵ✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + 1) tr♦♥❣ A✳ ✭①❡♠ √ ❍➻♥❤ ✾✮ ❱➻ |U | < 2λk = |fk | ♥➯♥ U ❝â t❤➸ ❦❤æ♥❣ ❤♦➔♥ t♦➔♥ ❝❤ù❛ ❝↔ Ik+1 ✈➔ Jk+1 ✳ ❚❛ ①➨t ❝→❝ ❦❤↔ ♥➠♥❣ s❛✉✳ ❑❤↔ ♥➠♥❣ ✶✳ ●✐↔ sû r➡♥❣ ❝↔ Ik+1 ✈➔ Jk+1 ✤➲✉ ❦❤ỉ♥❣ ❤♦➔♥ t♦➔♥ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ U õ tỗ t số ữỡ n1 , n2 s❛♦ ❝❤♦ U ∩ Ik+n1 = ∅, U ∩ Ik+n1 +1 = ∅, U ∩ Jk+n2 = ∅, U ∩ Jk+n2 +1 = ∅ ✸✸ ●✐↔ sû r➡♥❣ ❝❤♦ n1 ≥ ✈➔ n2 > 1✳ ▲➜② FUI = {fδ ∈ FU : fδ ⊂ Ik+1 } ✈➔ FUJ = {fδ ∈ FU : fδ ⊂ Jk+1 }✳ ❇➙② ❣✐í ✈➩ ❤❛✐ ✤÷í♥❣ l1 ✈➔ l2 ❝ị♥❣ s s ợ ữớ r ợ s ú ♥❤❛✉ ✈ỵ✐ ❜✐➯♥ U ✈➔ U ♥➠♠ ❣✐ú❛ ❝❤ó♥❣✳ ●å✐ ❦❤♦↔♥❣ ❝→❝❤ l1 ✈➔ Ai ❧➔ , (i = 1, 2) ✭①❡♠ ❍➻♥❤ ✾✮✳ ❑❤✐ ✤â✱ t❛ ❝â ≥ |λk+1 | − |λk+ni |, |fδ |s < (4n1 − 1)(|fk+n1 +1 |)s ✈➔ fδ ∈FUI |fδ |s < (4n2 − 1)(|fk+n2 +1 |)s ✳ fδ ∈FUI ❤➻♥❤ ✾✳ ❱➻ |U | > |fk | − 2|fk+1 | + a1 + a2 ✈➔ |fδ |s ≤ fδ ∈FU |fδ |s + fδ ∈FUI |fδ |s + |fk+1 |, fδ ∈FUJ ♥➯♥ ❦❤✐ λ ≤ 41 , n1 ≥ ✈➔ n2 > 1✱ t❤➻ t÷ì♥❣ tü ❝❤ù♥❣ ♠✐♥❤ ❚r÷í♥❣ ❤đ♣ ✷✱ t❛ ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ |U |s ≥ |fδ |s fδ ∈FU ●✐↔ sû r➡♥❣ n1 = n2 = 1✳ ◆➳✉ < a1 ≤ √ k+1 − λk+2 ) t❤➻ t❛ ❝â (λ ✸✹ √ k+1 (λ − λk+2 ) ✈➔ < a2 ≤ |U | > |fk | − 2|fk+1 | ✈➔ ❉♦ ✤â✱ ❦❤✐ λ ≤ t❛ ❝â |U | ≥ s √ fδ ∈FU fδ ∈FU |fδ |s ✳ |fδ |s ≤ 6(fk+2 )s ✳ √ − λ ) ✈➔ < a2 ≤ 22 (λk+1 − λk+2 )✱ t❛ ❝â ◆➳✉ a1 > √ |U |s > |fk | − 2|fk+1 | + a1 > 2λk (1 − 31 λ − 12 λ2 ) ✈➔ |fδ |s ≤ 8(fk+2 )s ✳ k+1 (λ ❉♦ ✤â✱ ❦❤✐ λ ≤ ◆➳✉ √ k+2 t❛ ❝â |U | ≥ a1 > 22 (λk+1 |U |s > |fk | ◆➯♥ ❦❤✐ λ ≤ fδ ∈FU |fδ | ✳ s s fδ ∈FU √ − λ ) ✈➔ a2 > 22 (λk+1 − λk+2 )✱ t❤➻ − 2|fk+1 | + a1 + a2 ✈➔ |fδ |s ≤ 8(fk+2 )s ✳ k+2 fδ ∈FU t❛ ❝â ✤÷đ❝ |U |s ≥ |fδ |s fδ ∈FU ❑❤↔ ♥➠♥❣ ✷✳ ●✐↔ sû r➡♥❣ Ik+1 ❤♦➦❝ Jk+1 ❤♦➔♥ t♦➔♥ ♥➡♠ tr♦♥❣ U ✳ ◆➳✉ Ik+1 ⊂ U ❦➨♦ t❤❡♦ a1 > |fk+1 | ✈➔ |fδ |s = |fk+1 |s fδ ∈FUI ❚r♦♥❣ ❦❤✐ ✤â✱ ♥➳✉ a2 ≥ |fk+1 | − |fk+n2 | ✈➔ ①↔② ①❛ t❤❡♦ ❦❤↔ ♥➠♥❣ ✶✱ t❛ ❝â |U | ≥ |fk | − 2|fk+1 | + a1 + a2 ✈➔ ❑❤✐ λ ≤ |fδ |s < (4n2 − 1)|fk+n2 +1 |s fδ ∈FUJ |fδ |s ≤ fδ ∈FU t❛ t❤✉ ✤÷đ❝ |U |s ≥ |fδ |s + 2|fk+1 | fδ ∈FUJ |fδ |s fδ ∈FU ❚r÷í♥❣ ❤đ♣ ✹✳ U ❣✐❛♦ ✈ỵ✐ ❜è♥ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + 1) tr♦♥❣❣ A✳ ❑❤↔ ♥➠♥❣ ✶✳ ●✐↔ sû U ❦❤æ♥❣ ❤♦➔♥ t♦➔♥ ❝❤ù❛ ❝→❝ t➟♣ Ik+1 , Ik+1 , Jk+1 ✈➔ Jk+1 t tỗ t số ữỡ ni (1 ≤ i ≤ 4)✱ s❛♦ ❝❤♦ U ∩ Ik+n1 = ∅, U ∩ Ik+n1 +1 = ∅, U ∩ Ik+n2 = ∅, U ∩ Ik+n2 +1 = ∅, U ∩ Jk+n3 = ∅, U ∩ Ik+n3 +1 = ∅, U ∩ Jk+n4 = ∅, U ∩ Jk+n4 +1 = ∅ ✸✺ ✣➦t U1 = U ∩ Ik+n1 , U2 = U ∩ Ik+n2 , U3 = U ∩ Jk+n3 ✱ ✈➔ U4 = U ∩ Jk+n4 ✱ t❛ ❦➼ ❤✐➺✉ FUI = {fδ ∈ FU : fδ ⊂ Ik+1 }, FUI = {fδ ∈ FU : fδ ⊂ Ik+1 }✱ FUJ = {fδ ∈ FU : fδ ⊂ Jk+1 } ✈➔ FUJ = {fδ ∈ FU : fδ ⊂ Jk+1 }✳ ❱➻ ♠é✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni ) ❝❤ù❛ ❜è♥ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + 1) ♥➯♥ t❤❡♦ ✷✳✽ t❤➻ U ❝â t❤➸ ❣✐❛♦ ✈ỵ✐ ♥❤✐➲✉ ♥❤➜t ❜❛ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + 1)✳ ❇➙② ❣✐í ✈➩ ❜è♥ ✤÷í♥❣ li , (1 ≤ i ≤ 4) s♦♥❣ s♦♥❣ ợ ữớ s ú U ✤â ❣✐❛♦ ✈ỵ✐ t➜t ❝↔ ❜✐➯♥ ❝õ❛ U ✳ ●å✐ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ li ✈➔ Ai ❧➔ , (1 ≤ i ≤ 4) ✭①❡♠ ❍➻♥❤ ✶✵✮✳ ●✐↔ sû r➡♥❣ ♠é✐ Ui , (1 ≤ i ≤ 4) ❣✐❛♦ ✈ỵ✐ ❝❤➾ ♠ët ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + 1) t❛ ❝â ≥ |fk+1 | − |fk+ni | ≥ |fk | + ✈➔ i=1 s (4ni − 3) + 2|fk+ni +1 |s |fδ | ≤ fδ ∈FU ❉♦ ✤â✱ ❦❤✐ λ ≤ i=1 t❛ t❤✉ ✤÷đ❝ |U |s ≥ |fδ |s fδ ∈FU ●✐↔ sû U1 ❝➢t ✈ỵ✐ ♥❤✐➲✉ ❤ì♥ ♠ët ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + 1) ✈➔ t➜t ❝↔ Ui , (2 ≤ i ≤ 4) ❣✐❛♦ ✈ỵ✐ ❝❤➾ ♠ët ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k+ni +1), (2 ≤ i ≤ 4)✳ tỗ t l s U1 ổ t❤➸ ❣✐❛♦ ✈ỵ✐ ❤❛✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + l) ♠ët tr♦♥❣ sè ✤â ❧➔ ð ♣❤➼❛ tr➯♥ ❜➯♥ ♣❤↔✐ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + 1) ❝→✐ ♠➔ ♣❤➼❛ tr➯♥ ❜➯♥ ♣❤↔✐ ❝õ❛ Ik+n1 ✳ ❚❤➟t ✈➟②✱ ♥➳✉ l t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr➯♥ t❤➻ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ i ≥ l✳ ❈❤ó♥❣ t❛ ❝❤å♥ l ❧➔ sè tè✐ t❤✐➸✉ ❝→❝ l ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ❝â a1 ≥ |fk+1 | − (|fk+n1 | − |fk+n1 +1 | + |fk+n1 +l−1 |) ✸✻ ✈➔ s (4n1 − 1)|fk+n1 +1 |s − 2|fk+n1 −l |s |fδ | < i=1 fδ ∈FUI ❤➻♥❤ ✶✵✳ ❱➻ ✈➟②✱ ❦❤✐ λ ≤ 41 ✱ t❛ t❤✉ ✤÷đ❝ |U |s ≥ |fδ |s fδ ∈FU ◆➳✉ l ♥❤÷ ✈➟② ❦❤ỉ♥❣ tỗ t t t õ |fk+1 | 21 (|fk+n1 | − |fk+n1 +1 |) ✈➔ ❉♦ ✤â✱ t❛ ❝â |fδ |s ≤ ✈➔ (4n1 − 1)|fk+n1 +1 |s ✳ i=1 fδ ∈FUI |U | ≥ |fk | − 2|fk+1 | + 4 i=1 s (4ni − 3)|fk+ni +1 |s + |fδ | ≤ fδ ∈FUI i=1 |fδ |s fδ ∈FUI ❚÷ì♥❣ tü ♥❤÷ ✈➟②✱ ❦❤✐ λ ≤ 41 ✱ t❛ ❝â |U |s ≥ |fδ |s fδ ∈FU ✸✼ ❑❤✐ ❜➜t ❦➻ ❤❛✐✱ ❜❛ ❤♦➦❝ ❜è♥ tr♦♥❣ Ui , ≤ i ≤ ❣✐❛♦ ✈ỵ✐ ♥❤✐➲✉ ❤ì♥ ♠ët ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð t❤ù (k + ni + 1), (1 ≤ i ≤ 4) t❤➻ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❧➔ ❤♦➔♥ t tữỡ tỹ ỵ tr ●✐↔ sû r➡♥❣ U ❤♦➔♥ t♦➔♥ ❝❤ù❛ Ik+1 ✈➔ Ik+1 , Jk+1 ✈➔ Jk+1 ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â a1 > fk+1 ✈➔ |fδ |s = |fk+1 |s fδ ∈FUI P❤➛♥ ❝á♥ ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ❑❤↔ ♥➠♥❣ ✶ ✈➔ ❞➵ ❞➔♥❣ ❝â ✤÷đ❝ ❦➳t q✉↔ ♠♦♥❣ ♠✉è♥✳ ❑❤↔ ♥➠♥❣ ✸✳ ●✐↔ sû r➡♥❣ U ❤♦➔♥ t♦➔♥ ❝❤ù❛ ❝↔ Ik+1 ✈➔ Ik+1 ✈➔ ❣✐❛♦ ✈ỵ✐ Jk+1 ✈➔ Jk+1 ✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ♥➔②✱ t÷ì♥❣ tü ❝❤ó♥❣ t❛ ❝â |U |s ≥ |fδ |s ✳ fδ ∈FU ❱➟② ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷✳✷ ❈ỉ♥❣ t❤ù❝ t➼♥❤ ✤ë ✤♦ ❝õ❛ t❤↔♠ ❙✐❡r♣✐♥s❦✐ ❚ø ❝→❝ ❇ê ✤➲ ❜ê trđ tr♦♥❣ ▼ư❝ ✷✳✷✳✶✱ t❛ t❤✉ ✤÷đ❝ ❦➳t q✉↔ s❛✉✳ ✭❬✸❪✮✳ ◆➳✉ λ ≤ 14 ✱ t❤➻ Hs (E) = HFs (E)✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {Ui } ❧➔ ♠ët tũ ỵ E ố ợ t U t tỗ t k > s❛♦ ❝❤♦ 2λk+1 ≤ Ui ≤ 2λk ✳ ●✐↔ sû FUi ❧➔ ❧ỵ♣ t➜t ❝↔ ❝→❝ ❤➻♥❤ ✈✉ỉ♥❣ ❝ì sð ❤♦➔♥ t♦➔♥ ❝❤ù❛ tr♦♥❣ Ui ❦❤ỉ♥❣ ❝→✐ ♥➔♦ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❝→✐ ❦❤→❝✳ ❚ø ❇ê ✤➲ ✷✳✷✳✶✳✷✱ t❤➻ |U |s ≥ |fδ |s ✤ó♥❣ ✈ỵ✐ λ ≤ 41 ✳ ❱➻ E ❧➔ t➟♣ ✷✳✷✳✷✳✶ ✣à♥❤ ❧➼ fδ ∈FU ❝♦♠♣❛❝t✱ ✈➻ ✈➟② ❝❤ó♥❣ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ FUi t❤ä❛ ♠➣♥ ❱➻ |Ui |s ≥ i s HF (E) i fδ ∈FU i fδ ∈FUi fδ ⊃ E ✳ |f |s Hs (E) HFs (E) ú ỗ t❤í✐✱ Hs (E) ≤ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❱➟② t❛ ❝â Hs (E) = HFs (E)✳ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✶✳✶ ✈➔ ✣à♥❤ ❧➼ ✷✳✷✳✷✳✶ t❛ ❝â ❍➺ q✉↔ s❛✉✳ s ✭❬✹❪✮✳ ◆➳✉ λ ≤ 41 ✱ t❤➻ Hs = 2 ✳ ✷✳✷✳✷✳✷ ❍➺ q✉↔ ✸✽ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉ ✶✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ✤ë ✤♦ ❍❛✉s❞♦r❢❢✱ ▼ỉ t↔ ❝→❝❤ ①➙② ❞ü♥❣ ❝ơ♥❣ ♥❤÷ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ ❙✐❡r♣✐♥s❦✐✳ ✷✳ ❍➺ t❤è♥❣ ❝→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ✈➔ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝→❝ ❝❤ù♥❣ ♠✐♥❤ tt t ữợ t tr ữợ sr t ✤➺♠ ❙✐❡r♣✐♥s❦✐ ✭▼ư❝ ✷✳✶ ✈➔ ✷✳✷ tr♦♥❣ ❈❤÷ì♥❣ ✷✮✳ ✸✳ ự tt ỵ ỵ ❑❍❷❖ ❬✶❪ ❊✳ ❆②❡r✱ ❘✳ ❙✳ ❛♥❞ ❙tr✐❝❤❛rt③✱✭✶✾✾✾✮ ❊①❛❝t ❍❛✉s❞♦r❢❢ ▼❡❛s✉r❡ ❛♥❞ ■♥✲ t❡r✈❛❧s ♦❢ ▼❛①✐♠✉♠ ❉❡♥s✐t② ❢♦r ❈❛♥t♦r ❙❡ts✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✺✶✱ ✸✼✷✺✲✸✼✹✶✳ ❬✷❪ ✳ ❏✳ ❇❛♦❣✉♦✱ ❩✳ ❩❤✐✇❡✐ ❛♥❞ ❩✳ ❩✉♦❧✐♥❣ ✭✷✵✵✶✮✱ ❍❛✉s❞♦r❢❢ ▼❡❛s✉r❡ ♦❢ ❙✐❡r♣✐♥s❦✐ ●❛s❦❡t ❛♥❞ ❙❡❧❢✲Pr♦❞✉❝t ❙❡ts ♦❢ ❈❛♥t♦r ❙❡ts ✭✐♥ ❈❤✐♥❡s❡✮✱ ❩❤♦♥❣s❤❛♥ ❯♥✐✈❡rs✐t②✱ ♣r❡♣r✐♥t✳ ❬✸❪ ❑✳ ❋❛❧❝♦♥❡r ✭✶✾✾✵✮✱ ❋r❛❝t❛❧ ●❡♦♠❡tr②✿ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦✉♥❞❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ◆❡✇ ❨♦r❦✿ ❏♦❤♥ ❲✐❧❡② & ❙♦♥s✳ ❬✹❪ ❈✳ ❍✉❛♥❣ ✭✷✵✵✵✮✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ❢♦r t❤❡ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡ ♦❢ ❙✐❡r♣✐♥s❦✐ ❝❛r♣❡t✱ ❆❝t❛ ▼❛t❤✳ ❙✐♥✐❝❛✱ ✹✸✱ ✺✾✾✲✻✵✹✳ ❬✺❪ ❏✳ ▼❛r✐♦♥ ✭✶✾✽✼✮✱ ▼❡❛s✉r❡ ❞❡ ❍❛✉s❞♦r❢❢ ❉✬❡♥s❡♠❜❧❡s ❋r❛❝t❛❧s✱ ❆♥♥✳ ❙❝✐✳ ▼❛t❤✳ ◗✉❡❜❡❝✱ ✶✶✱ ✶✶✶✲✶✸✼✳ ❬✻❪ ❩✳ ▲✳ ❩❤♦✉ ✭✶✾✾✼✮✱ ❚❤❡ ❍❛✉s❞♦r❢❢ ▼❡❛s✉r❡s ♦❢ t❤❡ ❑♦❝❤ ❈❝✉r✈❡ ❛♥❞ ❙✐❡r♣✐♥s❦✐ ●❛s❦❡t ✭✐♥ ❈❤✐♥❡s❡✮✱ Pr♦❣✳ ◆❛t✳ ❙❝✐✳✱ ✼✱ ✹✵✸✲✹✵✾✳ ❬✼❪ ❩✳ ▲✳ ❩❤♦✉ ✭✶✾✾✼✮✱ ❍❛✉s❞♦r❢❢ ▼❡❛s✉r❡ ♦❢ ❙✐❡r♣✐♥s❦✐ ●❛s❦❡t✱ ❙❝✐✳ ❈❤✐♥❛ ✭❙❡r✐❡s ❆✮✱ ✹✵✱ ✶✵✶✻✲✶✵✷✶✳ ❬✽❪ ❩✳ ▲✳ ❩❤♦✉ ❛♥❞ ▲✳ ❋❡♥❣ ✭✷✵✵✵✮✱ ❆ ◆❡✇ ❊❡st✐♠❛t❡ ♦❢ t❤❡ ❍❛✉s❞♦r❢❢ ▼❡❛✲ s✉r❡ ♦❢ t❤❡ ❙✐❡r♣✐♥s❦✐ ●❛s❦❡t✱ ◆♦♥❧✐♥❡❛r✐t②✱ ✶✸✱ ✹✼✾✲✹✾✶✳ ❆♣♣❧✐❝❛t✐♦♥s✱ ❲✐❧❡②✱ ◆❡✇ ❨♦r❦✳ ✹✵ ❬✾❪ ❩✳ ❩❤♦✉ ❛♥❞ ▼✳ ❲✉ ✭✶✾✾✾✮✱ ❚❤❡ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡ ♦❢ ❛ ❙✐❡r♣✐♥s❦✐ ❝❛r♣❡t✱ ❙❝✐✳ ❈❤✐♥❛ ❙❡r✳ ❆✱ ✹✼✱ ✻✼✸✲✻✽✵✳ ❬✶✵❪ ❳✳ ❍✳ ❲❛♥❣ ✭✶✾✾✾✮✱ ❊st✐♠❛t✐♦♥ ❛♥❞ ❈❝♦♥❥❡❝t✉r❡ ♦❢ t❤❡ ❍❛✉s❞♦r❢❢ ▼❡❛✲ s✉r❡ ♦❢ ❙✐❡r♣✐♥s❦✐ ●❛s❦❡t ✭✐♥ ❈❤✐♥❡s❡✮✱ Pr♦❣✳ ◆❛t✳ ❙❝✐✳✱ ✾✱ ✹✽✽✲✹✾✸✳ ✹✶