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✧❆ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ t❤✐♥❣s t❤❛t s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳✧ ❖❦✳✳✳ ❜✉t ❤♦✇ ❝❧♦s❡ ❝❛♥ t❤❡ ❛r❡❛s ❜❡❄

❖❦✳✳✳ ❜✉t ❤♦✇ ❝❧♦s❡ ❝❛♥ t❤❡ ❛r❡❛s ❜❡❄ ✧❆ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ t❤✐♥❣s t❤❛t s❤♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳✧

s❡❡♠s t❤❡ ❜❡st ♣♦ss✐❜❧❡ ■♥t✉✐t✐♦♥ ♦❢ ❧♦✇ ❞✐s❝r❡♣❛♥❝② s❡❡♠s ♥♦t ♦♣t✐♠❛❧

■♥t✉✐t✐♦♥ ♦❢ ❧♦✇ ❞✐s❝r❡♣❛♥❝② s❡❡♠s ♥♦t ♦♣t✐♠❛❧ s❡❡♠s t❤❡ ❜❡st ♣♦ss✐❜❧❡

❘ ❘▼❙ ❘ ✷ ▼❡❛s✉r✐♥❣ ❛r❡❛ ❞❡✈✐❛t✐♦♥ D ✿ ❞✐ss❡❝t✐♦♥ ✇✐t❤ tr✐❛♥❣❧❡ ❛r❡❛s A ✶ , . . . , A n ◮ ❘♦♦t✲♠❡❛♥✲sq✉❛r❡ ❡rr♦r ✭❘▼❙✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✮✿ � � n � � ✷ � ✶ A i − ✶ � � ❘▼❙ ( D ) := n n i = ✶ ◮ ❘❛♥❣❡✿ ❘ ( D ) = max i , j ∈ [ n ] | A i − A j |

▼❡❛s✉r✐♥❣ ❛r❡❛ ❞❡✈✐❛t✐♦♥ D ✿ ❞✐ss❡❝t✐♦♥ ✇✐t❤ tr✐❛♥❣❧❡ ❛r❡❛s A ✶ , . . . , A n ◮ ❘♦♦t✲♠❡❛♥✲sq✉❛r❡ ❡rr♦r ✭❘▼❙✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✮✿ � � n � � ✷ � ✶ A i − ✶ � � ❘▼❙ ( D ) := n n i = ✶ ◮ ❘❛♥❣❡✿ ❘ ( D ) = max i , j ∈ [ n ] | A i − A j | ❘ ( D ) ✷ √ n ≤ ❘▼❙ ( D ) ≤ ❘ ( D )

✹ ✸ ✷ ✶ ✶ ✷ ✶ ●r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭●r❛♣❤ Γ ♦❢ ❛ ❞✐ss❡❝t✐♦♥✮ ◆♦❞❡s ✿ ❝♦r♥❡rs ♦❢ tr✐❛♥❣❧❡s ❊❞❣❡ ✿ ❜❡t✇❡❡♥ ❝♦r♥❡rs ♦❢ ❛ tr✐❛♥❣❧❡ ♥♦t ❝♦♥t❛✐♥✐♥❣ s✐❞❡ ♥♦❞❡s ❉✐ss❡❝t✐♦♥

●r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭●r❛♣❤ Γ ♦❢ ❛ ❞✐ss❡❝t✐♦♥✮ ◆♦❞❡s ✿ ❝♦r♥❡rs ♦❢ tr✐❛♥❣❧❡s ❊❞❣❡ ✿ ❜❡t✇❡❡♥ ❝♦r♥❡rs ♦❢ ❛ tr✐❛♥❣❧❡ ♥♦t ❝♦♥t❛✐♥✐♥❣ s✐❞❡ ♥♦❞❡s c ✹ c ✸ i ✷ i ✶ c ✶ c ✷ b ✶ ◆♦❞❡s

✷ ✶ ●r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭●r❛♣❤ Γ ♦❢ ❛ ❞✐ss❡❝t✐♦♥✮ ◆♦❞❡s ✿ ❝♦r♥❡rs ♦❢ tr✐❛♥❣❧❡s ❊❞❣❡ ✿ ❜❡t✇❡❡♥ ❝♦r♥❡rs ♦❢ ❛ tr✐❛♥❣❧❡ ♥♦t ❝♦♥t❛✐♥✐♥❣ s✐❞❡ ♥♦❞❡s c ✹ c ✸ c ✶ c ✷ b ✶ ❇♦✉♥❞❛r② ♥♦❞❡s

✷ ✶ ✶ ●r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭●r❛♣❤ Γ ♦❢ ❛ ❞✐ss❡❝t✐♦♥✮ ◆♦❞❡s ✿ ❝♦r♥❡rs ♦❢ tr✐❛♥❣❧❡s ❊❞❣❡ ✿ ❜❡t✇❡❡♥ ❝♦r♥❡rs ♦❢ ❛ tr✐❛♥❣❧❡ ♥♦t ❝♦♥t❛✐♥✐♥❣ s✐❞❡ ♥♦❞❡s c ✹ c ✸ c ✶ c ✷ ❈♦r♥❡r ♥♦❞❡s

✹ ✸ ✶ ✷ ●r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭●r❛♣❤ Γ ♦❢ ❛ ❞✐ss❡❝t✐♦♥✮ ◆♦❞❡s ✿ ❝♦r♥❡rs ♦❢ tr✐❛♥❣❧❡s ❊❞❣❡ ✿ ❜❡t✇❡❡♥ ❝♦r♥❡rs ♦❢ ❛ tr✐❛♥❣❧❡ ♥♦t ❝♦♥t❛✐♥✐♥❣ s✐❞❡ ♥♦❞❡s i ✷ i ✶ b ✶ ❙✐❞❡ ♥♦❞❡s

●r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭●r❛♣❤ Γ ♦❢ ❛ ❞✐ss❡❝t✐♦♥✮ ◆♦❞❡s ✿ ❝♦r♥❡rs ♦❢ tr✐❛♥❣❧❡s ❊❞❣❡ ✿ ❜❡t✇❡❡♥ ❝♦r♥❡rs ♦❢ ❛ tr✐❛♥❣❧❡ ♥♦t ❝♦♥t❛✐♥✐♥❣ s✐❞❡ ♥♦❞❡s c ✹ c ✸ i ✷ i ✶ c ✶ c ✷ b ✶ ❊❞❣❡s

❙✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❙✐❞❡ ♥♦❞❡s − → ❧✐♥❡❛r ❝♦♥str❛✐♥ts i ✷ i ✶ b ✶ ❙✐❞❡ ♥♦❞❡s

✷ ✶ ❙✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❙✐❞❡ ♥♦❞❡s − → ❧✐♥❡❛r ❝♦♥str❛✐♥ts b ✶ b ✶ − → ( b ✶ , ( c ✶ , c ✷ ))

✷ ✶ ❙✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❙✐❞❡ ♥♦❞❡s − → ❧✐♥❡❛r ❝♦♥str❛✐♥ts i ✶ i ✶ − → ( i ✶ , ( b ✶ , i ✷ ))

✶ ✶ ❙✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❙✐❞❡ ♥♦❞❡s − → ❧✐♥❡❛r ❝♦♥str❛✐♥ts i ✷ i ✷ − → ( i ✷ , ( b ✶ , c ✸ ))

✷ ✶ ✶ ❙✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥ ❙✐❞❡ ♥♦❞❡s − → ❧✐♥❡❛r ❝♦♥str❛✐♥ts ❆♥♦t❤❡r ♣❧❛♥❛r ❣r❛♣❤ ✇✐t❤ ♠♦r❡ tr✐❛♥❣❧❡s✿ ❛ s✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❋r❛♠❡❞ ♠❛♣✮ ✷ t❤❛t s❡♥❞s t❤❡ ❝♦r♥❡r ♥♦❞❡s ❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❛ ♠❛♣ ♦❢ t♦ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡✳ ❉❡✜♥✐t✐♦♥ ✭❈♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❝♦♥str❛✐♥❡❞ ✐❢ s❡♥❞s t❤❡ s✐❞❡ ♥♦❞❡s ❛♥❞ t❤❡ t✇♦ ❝♦r♥❡rs ♦❢ t❤❛t s✐❞❡ t♦ ❛ ❧✐♥❡✱ ❢♦r ❡✈❡r② s✐❞❡ ♥♦❞❡✳ ❋r❛♠❡❞ ♠❛♣s V (Γ) := ♥♦❞❡s ♦❢ t❤❡ ❣r❛♣❤ Γ ♦❢ ❛ ✜①❡❞ ❞✐ss❡❝t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❈♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❝♦♥str❛✐♥❡❞ ✐❢ s❡♥❞s t❤❡ s✐❞❡ ♥♦❞❡s ❛♥❞ t❤❡ t✇♦ ❝♦r♥❡rs ♦❢ t❤❛t s✐❞❡ t♦ ❛ ❧✐♥❡✱ ❢♦r ❡✈❡r② s✐❞❡ ♥♦❞❡✳ ❋r❛♠❡❞ ♠❛♣s V (Γ) := ♥♦❞❡s ♦❢ t❤❡ ❣r❛♣❤ Γ ♦❢ ❛ ✜①❡❞ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭❋r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❛ ♠❛♣ φ : V (Γ) → R ✷ t❤❛t s❡♥❞s t❤❡ ❝♦r♥❡r ♥♦❞❡s ♦❢ Γ t♦ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡✳

❋r❛♠❡❞ ♠❛♣s V (Γ) := ♥♦❞❡s ♦❢ t❤❡ ❣r❛♣❤ Γ ♦❢ ❛ ✜①❡❞ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭❋r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❛ ♠❛♣ φ : V (Γ) → R ✷ t❤❛t s❡♥❞s t❤❡ ❝♦r♥❡r ♥♦❞❡s ♦❢ Γ t♦ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡✳ ❉❡✜♥✐t✐♦♥ ✭❈♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ φ ✐s ❝♦♥str❛✐♥❡❞ ✐❢ φ s❡♥❞s t❤❡ s✐❞❡ ♥♦❞❡s ❛♥❞ t❤❡ t✇♦ ❝♦r♥❡rs ♦❢ t❤❛t s✐❞❡ t♦ ❛ ❧✐♥❡✱ ❢♦r ❡✈❡r② s✐❞❡ ♥♦❞❡✳

❋r❛♠❡❞ ♠❛♣s V (Γ) := ♥♦❞❡s ♦❢ t❤❡ ❣r❛♣❤ Γ ♦❢ ❛ ✜①❡❞ ❞✐ss❡❝t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭❋r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❛ ♠❛♣ φ : V (Γ) → R ✷ t❤❛t s❡♥❞s t❤❡ ❝♦r♥❡r ♥♦❞❡s ♦❢ Γ t♦ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡✳ ❉❡✜♥✐t✐♦♥ ✭❈♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✮ ❆ ❢r❛♠❡❞ ♠❛♣ φ ✐s ❝♦♥str❛✐♥❡❞ ✐❢ φ s❡♥❞s t❤❡ s✐❞❡ ♥♦❞❡s ❛♥❞ t❤❡ t✇♦ ❝♦r♥❡rs ♦❢ t❤❛t s✐❞❡ t♦ ❛ ❧✐♥❡✱ ❢♦r ❡✈❡r② s✐❞❡ ♥♦❞❡✳ e d c d c d c d c e e e a a a a b b b b

❉❡✜♥✐t✐♦♥ ✭❆r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧✮ ❚❤❡ ❛r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ✐s t❤❡ ♣♦❧②♥♦♠✐❛❧ ✶ ✷ ✷ ✷ ✷ ✇❤❡r❡ ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡ ❛♥❞ ❞❡♥♦t❡s t❤❡ ❛r❡❛ ♦❢ tr✐❛♥❣❧❡ ✱ ✐✳❡✳ ❛ ❞❡t❡r♠✐♥❛♥t ♦❢ s✐③❡ ✸ ✶ ✶ ✶ ✶ ✶ ✷ ✸ ✷ ✶ ✷ ✸ ❍♦✇ t♦ ♠❡❛s✉r❡ ❞✐s❝r❡♣❛♥❝②❄ D = { t ✶ , t ✷ , . . . , t n } : ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s L : tr✐❛♥❣❧❡s ❢r♦♠ ❧✐♥❡❛r ❝♦♥str❛✐♥ts C = s❡t ♦❢ ❝♦r♥❡r ♥♦❞❡s Γ D

❛♥❞ ❞❡♥♦t❡s t❤❡ ❛r❡❛ ♦❢ tr✐❛♥❣❧❡ ✱ ✐✳❡✳ ❛ ❞❡t❡r♠✐♥❛♥t ♦❢ s✐③❡ ✸ ✶ ✶ ✶ ✶ ✶ ✷ ✸ ✷ ✶ ✷ ✸ ❍♦✇ t♦ ♠❡❛s✉r❡ ❞✐s❝r❡♣❛♥❝②❄ D = { t ✶ , t ✷ , . . . , t n } : ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s L : tr✐❛♥❣❧❡s ❢r♦♠ ❧✐♥❡❛r ❝♦♥str❛✐♥ts C = s❡t ♦❢ ❝♦r♥❡r ♥♦❞❡s Γ D ❉❡✜♥✐t✐♦♥ ✭❆r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧✮ ❚❤❡ ❛r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧ π D ∈ R [ X D ] ♦❢ D ✐s t❤❡ ♣♦❧②♥♦♠✐❛❧ � ✷ + � π D ( X D ) = � � ℓ ∈ L A ( ℓ ) ✷ + � � ( x v − p v ) ✷ + ( y v − q v ) ✷ � A ( t i ) − ✶ . i ∈ [ n ] n v ∈ C ✇❤❡r❡ ( p c , q c ) ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡

❍♦✇ t♦ ♠❡❛s✉r❡ ❞✐s❝r❡♣❛♥❝②❄ D = { t ✶ , t ✷ , . . . , t n } : ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s L : tr✐❛♥❣❧❡s ❢r♦♠ ❧✐♥❡❛r ❝♦♥str❛✐♥ts C = s❡t ♦❢ ❝♦r♥❡r ♥♦❞❡s Γ D ❉❡✜♥✐t✐♦♥ ✭❆r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧✮ ❚❤❡ ❛r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧ π D ∈ R [ X D ] ♦❢ D ✐s t❤❡ ♣♦❧②♥♦♠✐❛❧ � ✷ + � π D ( X D ) = � � ℓ ∈ L A ( ℓ ) ✷ + � � ( x v − p v ) ✷ + ( y v − q v ) ✷ � A ( t i ) − ✶ . i ∈ [ n ] n v ∈ C ✇❤❡r❡ ( p c , q c ) ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡ ❛♥❞ A ( t i ) ❞❡♥♦t❡s t❤❡ ❛r❡❛ ♦❢ tr✐❛♥❣❧❡ t i ✱ ✐✳❡✳ ❛ ❞❡t❡r♠✐♥❛♥t ♦❢ s✐③❡ ✸ � � ✶ ✶ ✶ � � A ( t i ) = ✶ � � x ✶ x ✷ x ✸ � � ✷ � � y ✶ y ✷ y ✸ � �

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ ✵ ✶ ✷ ✷ ✷ ✷ ❊①❛♠♣❧❡

■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ ✵ ✶ ✷ ✷ ✷ ✷ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ s ✹ • s ✸ • a • � ✷ = ✹✼ / ✶✹✹✵ � � A φ ( t i ) − ✶ t i ∈ D n • b • c s ✶ • s ✷ •

■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ ✵ ✶ ✷ ✷ ✷ ✷ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ s ✹ • s ✸ • a • � ✷ = ✹✼ / ✶✹✹✵ � � A φ ( t i ) − ✶ t i ∈ D n • b • c s ✶ • s ✷ •

■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ ✵ ✶ ✷ ✷ ✷ ✷ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ s ✹ • s ✸ • a • • b � ✷ = ✵ � � A φ ( t i ) − ✶ t i ∈ D n c • s ✶ • s ✷ •

■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ ✵ ✶ ✷ ✷ ✷ ✷ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ s ✹ • s ✸ • a • ✶ • b � ✷ = ✵ � � ✺ A φ ( t i ) − ✶ t i ∈ D n ✶ c • ✺ ✶ ✺ s ✶ • s ✷ •

■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ ✵ ✶ ✷ ✷ ✷ ✷ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ s ✹ • s ✸ • ✶ ✺ a • • b � ✷ = ✵ ✶ � � A φ ( t i ) − ✶ ✺ t i ∈ D n c • s ✶ • s ✷ •

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿ s ✹ • s ✸ • a • • b � ✷ = ✵ � � A φ ( t i ) − ✶ t i ∈ D n c • s ✶ • s ✷ • ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿ > ✵ � �� � � ✷ + � � A φ ( ℓ ) ✷ + � � ( x φ ( v ) − p v ) ✷ + ( y φ ( v ) − q v ) ✷ � π D ( X D ) = � A φ ( t i ) − ✶ . t i ∈ D n v ∈ C ℓ ∈ L

✵ ❞❡s❝r✐❜❡s ❛ ❝♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✱ ❛♥❞ ❛❧❧ s✐❣♥❡❞ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s ♦❢ ❛r❡ ❡q✉❛❧ t♦ ✶ ✳ ▼♦♥s❦②✬s t❤❡♦r❡♠ ✰ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥ ✵ ❛❧❧ ❞✐ss❡❝t✐♦♥s ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ tr✐❛♥❣❧❡s ✳ ❝♦♥str✳ ❢r❛♠❡❞ ♠❛♣ ♦❢ ❍♦✇ ❢❛st ❞♦❡s ✵ ❍♦✇ s♠❛❧❧ ❝❛♥ π D ( X D ) ❜❡❄ π D ( X D ) ≥ ✵ ❜② ❞❡✜♥✐t✐♦♥

▼♦♥s❦②✬s t❤❡♦r❡♠ ✰ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥ ✵ ❛❧❧ ❞✐ss❡❝t✐♦♥s ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ tr✐❛♥❣❧❡s ✳ ❝♦♥str✳ ❢r❛♠❡❞ ♠❛♣ ♦❢ ❍♦✇ ❢❛st ❞♦❡s ✵ ❍♦✇ s♠❛❧❧ ❝❛♥ π D ( X D ) ❜❡❄ π D ( X D ) ≥ ✵ ❜② ❞❡✜♥✐t✐♦♥ π D ( X D ) = ✵ ⇐ ⇒ X D ❞❡s❝r✐❜❡s ❛ ❝♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✱ ❛♥❞ ❛❧❧ s✐❣♥❡❞ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s ♦❢ D ❛r❡ ❡q✉❛❧ t♦ ✶ / n ✳

❛❧❧ ❞✐ss❡❝t✐♦♥s ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ tr✐❛♥❣❧❡s ✳ ❝♦♥str✳ ❢r❛♠❡❞ ♠❛♣ ♦❢ ❍♦✇ ❢❛st ❞♦❡s ✵ ❍♦✇ s♠❛❧❧ ❝❛♥ π D ( X D ) ❜❡❄ π D ( X D ) ≥ ✵ ❜② ❞❡✜♥✐t✐♦♥ π D ( X D ) = ✵ ⇐ ⇒ X D ❞❡s❝r✐❜❡s ❛ ❝♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✱ ❛♥❞ ❛❧❧ s✐❣♥❡❞ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s ♦❢ D ❛r❡ ❡q✉❛❧ t♦ ✶ / n ✳ ▼♦♥s❦②✬s t❤❡♦r❡♠ ✰ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥ = ⇒ π D ( X D ) > ✵

❍♦✇ ❢❛st ❞♦❡s ✵ ❍♦✇ s♠❛❧❧ ❝❛♥ π D ( X D ) ❜❡❄ π D ( X D ) ≥ ✵ ❜② ❞❡✜♥✐t✐♦♥ π D ( X D ) = ✵ ⇐ ⇒ X D ❞❡s❝r✐❜❡s ❛ ❝♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✱ ❛♥❞ ❛❧❧ s✐❣♥❡❞ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s ♦❢ D ❛r❡ ❡q✉❛❧ t♦ ✶ / n ✳ ▼♦♥s❦②✬s t❤❡♦r❡♠ ✰ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥ = ⇒ π D ( X D ) > ✵ D n := { ❛❧❧ ❞✐ss❡❝t✐♦♥s ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s } ✳ ∆( n ) := min { π D ( X φ ) | D ∈ D n , φ ❝♦♥str✳ ❢r❛♠❡❞ ♠❛♣ ♦❢ Γ D }

❍♦✇ s♠❛❧❧ ❝❛♥ π D ( X D ) ❜❡❄ π D ( X D ) ≥ ✵ ❜② ❞❡✜♥✐t✐♦♥ π D ( X D ) = ✵ ⇐ ⇒ X D ❞❡s❝r✐❜❡s ❛ ❝♦♥str❛✐♥❡❞ ❢r❛♠❡❞ ♠❛♣✱ ❛♥❞ ❛❧❧ s✐❣♥❡❞ ❛r❡❛s ♦❢ tr✐❛♥❣❧❡s ♦❢ D ❛r❡ ❡q✉❛❧ t♦ ✶ / n ✳ ▼♦♥s❦②✬s t❤❡♦r❡♠ ✰ s♦♠❡ ❝♦♠♣✉t❛t✐♦♥ = ⇒ π D ( X D ) > ✵ D n := { ❛❧❧ ❞✐ss❡❝t✐♦♥s ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s } ✳ ∆( n ) := min { π D ( X φ ) | D ∈ D n , φ ❝♦♥str✳ ❢r❛♠❡❞ ♠❛♣ ♦❢ Γ D } ❍♦✇ ❢❛st ❞♦❡s ∆( n ) n →∞ − → ✵ ?

✺ ✵ ✵✷✷✺ ✼ ✵ ✵✵✸✶ ✾ ✵ ✵✵✵✶✹ ✳ ✻ ✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ✶✶ ✹ ✷ ✶✵ ❞❡❝r❡❛s❡✳ Pr❡✈✐♦✉s r❡s✉❧ts ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❛♥❞ ❡①❤❛✉st✐✈❡ ❡♥✉♠❡r❛t✐♦♥ ▼❛♥s♦✇ ✭✷✵✵✸✮ ✉s❡❞ ▼❛t❧❛❜ t♦ st✉❞② t❤❡ r❛♥❣❡ ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s

✺ ✵ ✵✷✷✺ ✼ ✵ ✵✵✸✶ ✾ ✵ ✵✵✵✶✹ ✻ ✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ✶✶ ✹ ✷ ✶✵ ❞❡❝r❡❛s❡✳ Pr❡✈✐♦✉s r❡s✉❧ts ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❛♥❞ ❡①❤❛✉st✐✈❡ ❡♥✉♠❡r❛t✐♦♥ ▼❛♥s♦✇ ✭✷✵✵✸✮ ✉s❡❞ ▼❛t❧❛❜ t♦ st✉❞② t❤❡ r❛♥❣❡ ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s � � M ( n ) = min T ∈D n max t i , t j ∈ T | A ( t i ) − A ( t j ) | ✳

✼ ✵ ✵✵✸✶ ✾ ✵ ✵✵✵✶✹ ✻ ✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ✶✶ ✹ ✷ ✶✵ ❞❡❝r❡❛s❡✳ Pr❡✈✐♦✉s r❡s✉❧ts ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❛♥❞ ❡①❤❛✉st✐✈❡ ❡♥✉♠❡r❛t✐♦♥ ▼❛♥s♦✇ ✭✷✵✵✸✮ ✉s❡❞ ▼❛t❧❛❜ t♦ st✉❞② t❤❡ r❛♥❣❡ ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s � � M ( n ) = min T ∈D n max t i , t j ∈ T | A ( t i ) − A ( t j ) | ✳ M ( ✺ ) ≤ ✵ . ✵✷✷✺

✾ ✵ ✵✵✵✶✹ ✻ ✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ✶✶ ✹ ✷ ✶✵ ❞❡❝r❡❛s❡✳ Pr❡✈✐♦✉s r❡s✉❧ts ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❛♥❞ ❡①❤❛✉st✐✈❡ ❡♥✉♠❡r❛t✐♦♥ ▼❛♥s♦✇ ✭✷✵✵✸✮ ✉s❡❞ ▼❛t❧❛❜ t♦ st✉❞② t❤❡ r❛♥❣❡ ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s � � M ( n ) = min T ∈D n max t i , t j ∈ T | A ( t i ) − A ( t j ) | ✳ M ( ✺ ) ≤ ✵ . ✵✷✷✺ M ( ✼ ) ≤ ✵ . ✵✵✸✶

✻ ✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ✶✶ ✹ ✷ ✶✵ ❞❡❝r❡❛s❡✳ Pr❡✈✐♦✉s r❡s✉❧ts ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❛♥❞ ❡①❤❛✉st✐✈❡ ❡♥✉♠❡r❛t✐♦♥ ▼❛♥s♦✇ ✭✷✵✵✸✮ ✉s❡❞ ▼❛t❧❛❜ t♦ st✉❞② t❤❡ r❛♥❣❡ ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s � � M ( n ) = min T ∈D n max t i , t j ∈ T | A ( t i ) − A ( t j ) | ✳ M ( ✺ ) ≤ ✵ . ✵✷✷✺ M ( ✼ ) ≤ ✵ . ✵✵✸✶ M ( ✾ ) ≤ ✵ . ✵✵✵✶✹

Pr❡✈✐♦✉s r❡s✉❧ts ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❛♥❞ ❡①❤❛✉st✐✈❡ ❡♥✉♠❡r❛t✐♦♥ ▼❛♥s♦✇ ✭✷✵✵✸✮ ✉s❡❞ ▼❛t❧❛❜ t♦ st✉❞② t❤❡ r❛♥❣❡ ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s � � M ( n ) = min T ∈D n max t i , t j ∈ T | A ( t i ) − A ( t j ) | ✳ M ( ✺ ) ≤ ✵ . ✵✷✷✺ M ( ✼ ) ≤ ✵ . ✵✵✸✶ M ( ✾ ) ≤ ✵ . ✵✵✵✶✹ ❛♥❞ M ( ✶✶ ) ≤ ✹ . ✷ × ✶✵ − ✻ ✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝r❡❛s❡✳

❙❝❤✉❧③❡ ✭✷✵✶✶✮ ♦❜t❛✐♥❡❞ ❛ ❢❛♠✐❧② ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ r❛♥❣❡ ♦❢ ✸ ✳ ❛r❡❛ ❛t ♠♦st ✶ Pr♦♦❢ t❡❝❤♥✐q✉❡✿ ✉s❡❞ t❤❡ t❤❡♦r② ♦❢ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ Pr❡✈✐♦✉s r❡s✉❧ts ❯♣♣❡r ❜♦✉♥❞ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s ❊❛s② ❝♦♥str✉❝t✐♦♥s✿ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ ❢♦r♠ O ( ✶ / n ✷ ) ✳

Pr♦♦❢ t❡❝❤♥✐q✉❡✿ ✉s❡❞ t❤❡ t❤❡♦r② ♦❢ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳ Pr❡✈✐♦✉s r❡s✉❧ts ❯♣♣❡r ❜♦✉♥❞ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s ❊❛s② ❝♦♥str✉❝t✐♦♥s✿ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ ❢♦r♠ O ( ✶ / n ✷ ) ✳ ❙❝❤✉❧③❡ ✭✷✵✶✶✮ ♦❜t❛✐♥❡❞ ❛ ❢❛♠✐❧② ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ r❛♥❣❡ ♦❢ ❛r❡❛ ❛t ♠♦st O ( ✶ / n ✸ ) ✳

Pr❡✈✐♦✉s r❡s✉❧ts ❯♣♣❡r ❜♦✉♥❞ ❢♦r tr✐❛♥❣✉❧❛t✐♦♥s ❊❛s② ❝♦♥str✉❝t✐♦♥s✿ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ ❢♦r♠ O ( ✶ / n ✷ ) ✳ ❙❝❤✉❧③❡ ✭✷✵✶✶✮ ♦❜t❛✐♥❡❞ ❛ ❢❛♠✐❧② ♦❢ tr✐❛♥❣✉❧❛t✐♦♥s ✇✐t❤ r❛♥❣❡ ♦❢ ❛r❡❛ ❛t ♠♦st O ( ✶ / n ✸ ) ✳ Pr♦♦❢ t❡❝❤♥✐q✉❡✿ ✉s❡❞ t❤❡ t❤❡♦r② ♦❢ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳

Pr♦♦❢ t❡❝❤♥✐q✉❡✿ ●❛♣ t❤❡♦r❡♠s ❢r♦♠ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✳ ✏❆♥ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r ✵ ❝❛♥ ♥♦t ❜❡ ❛r❜✐tr❛r✐❧② ❝❧♦s❡ t♦ ✵✳✑ ✳✳✳❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❞❡❣r❡❡ ❛♥❞ t❤❡ s✐③❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ✐ts ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✳ ◆❡✇ r❡s✉❧ts ✕ ▲♦✇❡r ❜♦✉♥❞ ❇❡❝❛✉s❡ ❘ ( D ) ≥ ❘▼❙ ( D ) ❛♥❞ n ❘▼❙ ( D ) ✷ = π D ( X D ) ✱ ✐t s✉✣❝❡s t♦ ❣❡t ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r π D ( X D ) t♦ ❜♦✉♥❞ ❘ ( D ) ✳ ❲❡ ❣❡t ✶ ❘ ( D ) ≥ ✷ ✷ O ( n ) ✭❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✮

◆❡✇ r❡s✉❧ts ✕ ▲♦✇❡r ❜♦✉♥❞ ❇❡❝❛✉s❡ ❘ ( D ) ≥ ❘▼❙ ( D ) ❛♥❞ n ❘▼❙ ( D ) ✷ = π D ( X D ) ✱ ✐t s✉✣❝❡s t♦ ❣❡t ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r π D ( X D ) t♦ ❜♦✉♥❞ ❘ ( D ) ✳ ❲❡ ❣❡t ✶ ❘ ( D ) ≥ ✷ ✷ O ( n ) ✭❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✮ Pr♦♦❢ t❡❝❤♥✐q✉❡✿ ●❛♣ t❤❡♦r❡♠s ❢r♦♠ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✳ ✏❆♥ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r α � = ✵ ❝❛♥ ♥♦t ❜❡ ❛r❜✐tr❛r✐❧② ❝❧♦s❡ t♦ ✵✳✑ ✳✳✳❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❞❡❣r❡❡ ❛♥❞ t❤❡ s✐③❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ✐ts ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✳

❯s✐♥❣ ✐♥t✉✐t✐♦♥s ❢r♦♠ ❡①❤❛✉st✐✈❡ ❣❡♥❡r❛t✐♦♥ ❛♥❞ ❡①❝❡♣t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❢♦r ❡✈❡r② ♦❞❞ ✇✐t❤ ✶ ✶ ✭s✉♣❡r♣♦❧②♥♦♠✐❛❧✮ ❘ ✺ ✷ ✷ ✷ ◆❡✇ r❡s✉❧ts ✕ ❯♣♣❡r ❜♦✉♥❞ ❚♦ ♣r♦✈✐❞❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ✐t r❡q✉✐r❡s t♦ ❝♦♥str✉❝t ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✈❡r② s♠❛❧❧ r❛♥❣❡✳

✇❡ ♣r♦✈✐❞❡ ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❢♦r ❡✈❡r② ♦❞❞ ✇✐t❤ ✶ ✶ ✭s✉♣❡r♣♦❧②♥♦♠✐❛❧✮ ❘ ✺ ✷ ✷ ✷ ◆❡✇ r❡s✉❧ts ✕ ❯♣♣❡r ❜♦✉♥❞ ❚♦ ♣r♦✈✐❞❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ✐t r❡q✉✐r❡s t♦ ❝♦♥str✉❝t ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✈❡r② s♠❛❧❧ r❛♥❣❡✳ ❯s✐♥❣ ✐♥t✉✐t✐♦♥s ❢r♦♠ ❡①❤❛✉st✐✈❡ ❣❡♥❡r❛t✐♦♥ ❛♥❞ ❡①❝❡♣t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✱

◆❡✇ r❡s✉❧ts ✕ ❯♣♣❡r ❜♦✉♥❞ ❚♦ ♣r♦✈✐❞❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ✐t r❡q✉✐r❡s t♦ ❝♦♥str✉❝t ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✈❡r② s♠❛❧❧ r❛♥❣❡✳ ❯s✐♥❣ ✐♥t✉✐t✐♦♥s ❢r♦♠ ❡①❤❛✉st✐✈❡ ❣❡♥❡r❛t✐♦♥ ❛♥❞ ❡①❝❡♣t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✱ ✇❡ ♣r♦✈✐❞❡ ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s Z n ❢♦r ❡✈❡r② ♦❞❞ n ✇✐t❤ ✶ ✶ ❘ ( Z n ) ≤ n log ✷ n − ✺ = ✷ Ω(log ✷ n ) ✭s✉♣❡r♣♦❧②♥♦♠✐❛❧✮

✳✳✳ ❜❛❝❦ t♦ t❤❡ ❛r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧✳ ❍♦✇ t♦ ❣❡t ❛ ❧♦✇❡r ❜♦✉♥❞ ♦♥ π D ( X D ) ❄

t❤❡♥ t❤❡ ♠✐♥✐♠✉♠ ♦❢ ♦♥ t❤❡ ✲s✐♠♣❧❡① s❛t✐s✜❡s ✷ ✷ ✸ ✸ ✶ ✶ ✷ ✶ ✐s t❤❡ ❉❛✈❡♥♣♦rt✕▼❛❤❧❡r✕▼✐❣♥♦tt❡ ❜♦✉♥❞✳ ❆♥s❛t③✿ ❣❛♣ t❤❡♦r❡♠ ✐♥ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ❚❤❡♦r❡♠ ✭❊♠✐r✐s✕▼♦✉rr❛✐♥✕❚s✐❣❛r✐❞❛s✱ ✷✵✶✵✮ ■❢ f ∈ Z [ x ✶ , . . . , x k ] ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ♦♥ t❤❡ k ✲s✐♠♣❧❡①✿ � � k � x ∈ R k ≥ ✵ : x i ≤ ✶ , i = ✶ ❛♥❞ f ✐s ♦❢ ❞❡❣r❡❡ d ✱ ✇✐t❤ ❝♦❡✣❝✐❡♥ts ❜♦✉♥❞❡❞ ❜② ✷ τ ✱

❆♥s❛t③✿ ❣❛♣ t❤❡♦r❡♠ ✐♥ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ❚❤❡♦r❡♠ ✭❊♠✐r✐s✕▼♦✉rr❛✐♥✕❚s✐❣❛r✐❞❛s✱ ✷✵✶✵✮ ■❢ f ∈ Z [ x ✶ , . . . , x k ] ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ♦♥ t❤❡ k ✲s✐♠♣❧❡①✿ � � k � x ∈ R k ≥ ✵ : x i ≤ ✶ , i = ✶ ❛♥❞ f ✐s ♦❢ ❞❡❣r❡❡ d ✱ ✇✐t❤ ❝♦❡✣❝✐❡♥ts ❜♦✉♥❞❡❞ ❜② ✷ τ ✱ t❤❡♥ t❤❡ ♠✐♥✐♠✉♠ m DMM ♦❢ f ♦♥ t❤❡ k ✲s✐♠♣❧❡① s❛t✐s✜❡s √ � ( k ✷ + k ) log k ✷ log d + k ( ✸ + ✸ log d + τ + d log k ) − log m DMM < d + + d (log k + ✶ ) + log d + τ + ✷ ] d ( d − ✶ ) k − ✶ . m DMM ✐s t❤❡ ❉❛✈❡♥♣♦rt✕▼❛❤❧❡r✕▼✐❣♥♦tt❡ ❜♦✉♥❞✳

❆♥s❛t③✿ ❣❛♣ t❤❡♦r❡♠ ✐♥ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ❈♦r♦❧❧❛r② ❚❤❡ ♠✐♥✐♠✉♠ M ❢♦r t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧ π D ( X D ) s❛t✐s✜❡s − log M = O ( n ✷ ✾ n ) . ■♥ ♦t❤❡r ✇♦r❞s✱ ✶ ✶ ∆( n ) = ✷ O ( n ✷ ✾ n ) = ✷ ✷ O ( n ) .

❖♣❡♥ ◗✉❡st✐♦♥✿ ❍♦✇ t♦ ❣❡t ❛ ❜❡tt❡r ❧♦✇❡r ❜♦✉♥❞❄

❊①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ▼❛♥s♦✇ t♦ ❞✐ss❡❝t✐♦♥s ❋✐♥❞ ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ✉♣♣❡r ❜♦✉♥❞s ✶✳ ❯s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❧❛♥tr✐ ✱ ❛♥❞ ❙❛❣❡ t♦ ❣❡♥❡r❛t❡ ❛❧❧ ❞✐ss❡❝t✐♦♥s ✇✐t❤ tr✐❛♥❣❧❡s ❛♥❞ ✈❡rt✐❝❡s ✷✳ ❯s❡ ❇❡rt✐♥✐ ❛♥❞ s❝✐♣② t♦ ✜♥❞ ♦♣t✐♠❛ ❢♦r ❡❛❝❤ ❞✐ss❡❝t✐♦♥ ❆❜✉s❡ ❛♥❞ ❛✉t♦♠❛t✐③❡ ss❤ ✱ ❛♥❞ s❝r❡❡♥ ♦♥ ✸✻ ♣r♦❝❡ss♦rs ✐♥ t❤❡ ✐♥st✐t✉t❡✳ ●❡♥❡r❛t❡ ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✾ tr✐❛♥❣❧❡s ❛♥❞ ✽ ✈❡rt✐❝❡s t♦♦❦ ✸ ❞❛②s ■♠♣r♦✈✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❖r✐❣✐♥❛❧ ❣♦❛❧s✿

✶✳ ❯s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❧❛♥tr✐ ✱ ❛♥❞ ❙❛❣❡ t♦ ❣❡♥❡r❛t❡ ❛❧❧ ❞✐ss❡❝t✐♦♥s ✇✐t❤ tr✐❛♥❣❧❡s ❛♥❞ ✈❡rt✐❝❡s ✷✳ ❯s❡ ❇❡rt✐♥✐ ❛♥❞ s❝✐♣② t♦ ✜♥❞ ♦♣t✐♠❛ ❢♦r ❡❛❝❤ ❞✐ss❡❝t✐♦♥ ❆❜✉s❡ ❛♥❞ ❛✉t♦♠❛t✐③❡ ss❤ ✱ ❛♥❞ s❝r❡❡♥ ♦♥ ✸✻ ♣r♦❝❡ss♦rs ✐♥ t❤❡ ✐♥st✐t✉t❡✳ ●❡♥❡r❛t❡ ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✾ tr✐❛♥❣❧❡s ❛♥❞ ✽ ✈❡rt✐❝❡s t♦♦❦ ✸ ❞❛②s ■♠♣r♦✈✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❖r✐❣✐♥❛❧ ❣♦❛❧s✿ ◮ ❊①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ▼❛♥s♦✇ t♦ ❞✐ss❡❝t✐♦♥s ◮ ❋✐♥❞ ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ✉♣♣❡r ❜♦✉♥❞s

✷✳ ❯s❡ ❇❡rt✐♥✐ ❛♥❞ s❝✐♣② t♦ ✜♥❞ ♦♣t✐♠❛ ❢♦r ❡❛❝❤ ❞✐ss❡❝t✐♦♥ ❆❜✉s❡ ❛♥❞ ❛✉t♦♠❛t✐③❡ ss❤ ✱ ❛♥❞ s❝r❡❡♥ ♦♥ ✸✻ ♣r♦❝❡ss♦rs ✐♥ t❤❡ ✐♥st✐t✉t❡✳ ●❡♥❡r❛t❡ ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✾ tr✐❛♥❣❧❡s ❛♥❞ ✽ ✈❡rt✐❝❡s t♦♦❦ ✸ ❞❛②s ■♠♣r♦✈✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❖r✐❣✐♥❛❧ ❣♦❛❧s✿ ◮ ❊①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ▼❛♥s♦✇ t♦ ❞✐ss❡❝t✐♦♥s ◮ ❋✐♥❞ ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ✉♣♣❡r ❜♦✉♥❞s ✶✳ ❯s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❧❛♥tr✐ ✱ ❛♥❞ ❙❛❣❡ t♦ ❣❡♥❡r❛t❡ ❛❧❧ ❞✐ss❡❝t✐♦♥s ✇✐t❤ n tr✐❛♥❣❧❡s ❛♥❞ k ✈❡rt✐❝❡s

❆❜✉s❡ ❛♥❞ ❛✉t♦♠❛t✐③❡ ss❤ ✱ ❛♥❞ s❝r❡❡♥ ♦♥ ✸✻ ♣r♦❝❡ss♦rs ✐♥ t❤❡ ✐♥st✐t✉t❡✳ ●❡♥❡r❛t❡ ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✾ tr✐❛♥❣❧❡s ❛♥❞ ✽ ✈❡rt✐❝❡s t♦♦❦ ✸ ❞❛②s ■♠♣r♦✈✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❖r✐❣✐♥❛❧ ❣♦❛❧s✿ ◮ ❊①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ▼❛♥s♦✇ t♦ ❞✐ss❡❝t✐♦♥s ◮ ❋✐♥❞ ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ✉♣♣❡r ❜♦✉♥❞s ✶✳ ❯s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❧❛♥tr✐ ✱ ❛♥❞ ❙❛❣❡ t♦ ❣❡♥❡r❛t❡ ❛❧❧ ❞✐ss❡❝t✐♦♥s ✇✐t❤ n tr✐❛♥❣❧❡s ❛♥❞ k ✈❡rt✐❝❡s ✷✳ ❯s❡ ❇❡rt✐♥✐ ❛♥❞ s❝✐♣② t♦ ✜♥❞ ♦♣t✐♠❛ ❢♦r ❡❛❝❤ ❞✐ss❡❝t✐♦♥

●❡♥❡r❛t❡ ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✾ tr✐❛♥❣❧❡s ❛♥❞ ✽ ✈❡rt✐❝❡s t♦♦❦ ✸ ❞❛②s ■♠♣r♦✈✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❖r✐❣✐♥❛❧ ❣♦❛❧s✿ ◮ ❊①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ▼❛♥s♦✇ t♦ ❞✐ss❡❝t✐♦♥s ◮ ❋✐♥❞ ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ✉♣♣❡r ❜♦✉♥❞s ✶✳ ❯s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❧❛♥tr✐ ✱ ❛♥❞ ❙❛❣❡ t♦ ❣❡♥❡r❛t❡ ❛❧❧ ❞✐ss❡❝t✐♦♥s ✇✐t❤ n tr✐❛♥❣❧❡s ❛♥❞ k ✈❡rt✐❝❡s ✷✳ ❯s❡ ❇❡rt✐♥✐ ❛♥❞ s❝✐♣② t♦ ✜♥❞ ♦♣t✐♠❛ ❢♦r ❡❛❝❤ ❞✐ss❡❝t✐♦♥ � ❆❜✉s❡ ❛♥❞ ❛✉t♦♠❛t✐③❡ ss❤ ✱ ❛♥❞ s❝r❡❡♥ ♦♥ ✸✻ ♣r♦❝❡ss♦rs ✐♥ t❤❡ ✐♥st✐t✉t❡✳

■♠♣r♦✈✐♥❣ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❖r✐❣✐♥❛❧ ❣♦❛❧s✿ ◮ ❊①t❡♥❞ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ▼❛♥s♦✇ t♦ ❞✐ss❡❝t✐♦♥s ◮ ❋✐♥❞ ❣♦♦❞ ❝❛♥❞✐❞❛t❡s ❢♦r ✉♣♣❡r ❜♦✉♥❞s ✶✳ ❯s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣❧❛♥tr✐ ✱ ❛♥❞ ❙❛❣❡ t♦ ❣❡♥❡r❛t❡ ❛❧❧ ❞✐ss❡❝t✐♦♥s ✇✐t❤ n tr✐❛♥❣❧❡s ❛♥❞ k ✈❡rt✐❝❡s ✷✳ ❯s❡ ❇❡rt✐♥✐ ❛♥❞ s❝✐♣② t♦ ✜♥❞ ♦♣t✐♠❛ ❢♦r ❡❛❝❤ ❞✐ss❡❝t✐♦♥ � ❆❜✉s❡ ❛♥❞ ❛✉t♦♠❛t✐③❡ ss❤ ✱ ❛♥❞ s❝r❡❡♥ ♦♥ ✸✻ ♣r♦❝❡ss♦rs ✐♥ t❤❡ ✐♥st✐t✉t❡✳ ●❡♥❡r❛t❡ ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ❞✐ss❡❝t✐♦♥s ✇✐t❤ ✾ tr✐❛♥❣❧❡s ❛♥❞ ✽ ✈❡rt✐❝❡s t♦♦❦ ✸ ❞❛②s

❈❛♥ ❤❛✈❡ ❞✐♠❡♥s✐♦♥ ✵ ❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❞❡❣❡♥❡r❛t❡ ♦r ✢✐♣✲♦✈❡r ❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❤❛✈❡ ♠✐♥✐♠❛ ♦✉ts✐❞❡ t❤❡ sq✉❛r❡ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡✈✐❞❡♥❝❡s ❲❡ ♥♦✇ ❦♥♦✇ ♠♦r❡ ♦♥ t❤❡ ❣r❛❞✐❡♥t ✈❛r✐❡t②✿

❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❞❡❣❡♥❡r❛t❡ ♦r ✢✐♣✲♦✈❡r ❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❤❛✈❡ ♠✐♥✐♠❛ ♦✉ts✐❞❡ t❤❡ sq✉❛r❡ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡✈✐❞❡♥❝❡s ❲❡ ♥♦✇ ❦♥♦✇ ♠♦r❡ ♦♥ t❤❡ ❣r❛❞✐❡♥t ✈❛r✐❡t②✿ ◮ ❈❛♥ ❤❛✈❡ ❞✐♠❡♥s✐♦♥ > ✵

❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❤❛✈❡ ♠✐♥✐♠❛ ♦✉ts✐❞❡ t❤❡ sq✉❛r❡ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡✈✐❞❡♥❝❡s ❲❡ ♥♦✇ ❦♥♦✇ ♠♦r❡ ♦♥ t❤❡ ❣r❛❞✐❡♥t ✈❛r✐❡t②✿ ◮ ❈❛♥ ❤❛✈❡ ❞✐♠❡♥s✐♦♥ > ✵ ◮ ❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❞❡❣❡♥❡r❛t❡ ♦r ✢✐♣✲♦✈❡r

❈♦♠♣✉t❛t✐♦♥❛❧ ❡✈✐❞❡♥❝❡s ❲❡ ♥♦✇ ❦♥♦✇ ♠♦r❡ ♦♥ t❤❡ ❣r❛❞✐❡♥t ✈❛r✐❡t②✿ ◮ ❈❛♥ ❤❛✈❡ ❞✐♠❡♥s✐♦♥ > ✵ ◮ ❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❞❡❣❡♥❡r❛t❡ ♦r ✢✐♣✲♦✈❡r ◮ ❙♦♠❡ ❞✐ss❡❝t✐♦♥s ❤❛✈❡ ♠✐♥✐♠❛ ♦✉ts✐❞❡ t❤❡ sq✉❛r❡

❉✐ss❡❝t✐♦♥s ❛❝❤✐❡✈❡ ❜❡tt❡r ❜♦✉♥❞s ✼ tr✐❛♥❣❧❡s ❚r✐❛♥❣✉❧❛t✐♦♥s ❉✐ss❡❝t✐♦♥s π D ( X D ) ✵✳✵✵✵✵✶✶✹✹✸✸✷✻✽ ✵✳✵✵✵✶✽✸✸✸✵✽✾✶ ✼ ✈❡rt✐❝❡s ❘❛♥❣❡ ✵✳✵✵✹✵✵✽✶✵ ✵✳✵✶✷✼✽✼✾ ✹ . ✷✸✺✻✻✽✾✽ × ✶✵ − ✻ π D ( X D ) ✵✳✵✵✵✵✼✺✸✷✾✵ ✽ ✈❡rt✐❝❡s ❘❛♥❣❡ ✵✳✵✶✵✷✶✹✾ ✵✳✵✵✷✸✷✵✻✽ n ❘▼❙ ✶ . ✶✼✽✺✶ × ✶✵ − ✶ ✸ ✶ . ✵✷✾✺ × ✶✵ − ✷ ✺ ✼ . ✼✼✽✼✽✽ × ✶✵ − ✹ ✼ ✷ . ✼✸✻✽✸✾ × ✶✵ − ✹ ✾ ∗

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✺ ✳ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ❛ r❛♥❣❡ ♦r❞❡r ♦❢ ✶ ❆ ♥✐❝❡ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s

❆ ♥✐❝❡ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ❛ r❛♥❣❡ ♦r❞❡r ♦❢ O ( ✶ / n ✺ ) ✳

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st ✳ ✷ ❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ ✶ / n

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st ✳ ✷ ❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ + ✶ / n +

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st ✳ ✷ ❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ + , − ✶ / n + −

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st ✳ ✷ ❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ + , − , − , + ✶ / n + + − −

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st ✳ ✷ ❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ + , − , − , + , − , + , + , − , ✶ / n + + + + − − − −

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st ✳ ✷ ❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ + , − , − , + , − , + , + , − , ❡t❝✳ ✶ / n + + + + + + + + − − − − − − − −

❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ + , − , − , + , − , + , + , − , ❡t❝✳ ✶ / n + + + + + + + + − − − − − − − − ❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮ ✶ ❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ♠✐♥✐♠❛❧ r❛♥❣❡ ❛t ♠♦st n Ω(log ✷ n ) ✳

❊st✐♠❛t✐♥❣ t❤❡ ❡rr♦r S T 1 /n 2 /n F G R a 1 a 2 a 3 . . . a i a n − 1 a i +1 . . . Q O P E ❙❡t A i := a ✶ + · · · + a i ✱ ✇❡ ❤❛✈❡ △ EGO △ EFO = n / ✹ − A i + ✶ ❛♥❞ RO / SO = QO / PO n / ✹ − A i

❊st✐♠❛t✐♥❣ t❤❡ ❡rr♦r S T 1 /n 2 /n F G R a 1 a 2 a 3 . . . a i a n − 1 a i +1 . . . Q O P E ❚♦ ❡♥❞ ✇✐t❤ ❛ ✈❡rt✐❝❛❧ s❡❣♠❡♥t✱ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ r❛t✐♦s ♦❢ ✧✰✧ ❛♥❞ ✧✲✧ s❤♦✉❧❞ ❡q✉❛❧ RO / SO ❛♥❞ QO / PO ✿ n − ✶ � n / ✹ − A i + ✶ � τ i � ! = ✶ . n / ✹ − A i i = ✶

✷ ❛♥❞ ✇r✐t❡ ✶ ✶ ❙❡t ✹ ✶ ✶ ✶ ❚❛❦❡ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ ❛♥❞ ❡①♣r❡ss ✐t ❛s ❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ❛r♦✉♥❞ ✶ ❯s❡ t❤❡ ❧❡♠♠❛ t♦ ♠❛❦❡ t❤❡ ❛r❡❛s ✬s ❜❡ ❝❧♦s❡ t♦ ✶ t♦ ❛ ✏❤✐❣❤ ❞❡❣r❡❡✑ ❚❤❡ ❦❡② ♣r♦♣❡rt② ❚❤❡ ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡ { s i } i ≥ ✶ ❛♥♥✐❤✐❧❛t❡s ♣♦✇❡rs✿ ▲❡♠♠❛ ✭Pr♦✉❡t ✭✶✽✺✶✮✮ ▲❡t k ≥ ✵ ✱ b � = ✵ ✱ ❛♥❞ ❧❡t f ( x ) ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ d ✳ ■❢ d ≥ k ✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ F ( x ) ♦❢ ❞❡❣r❡❡ d − k s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t② ❤♦❧❞s ❢♦r ❛❧❧ x ✵ ✿ ✷ k � s i f ( x ✵ + ib ) = F ( x ✵ ) . i = ✶ ❖t❤❡r✇✐s❡✱ ✐❢ d < k ✱ t❤❡ ❛❜♦✈❡ s✉♠ ✐s ③❡r♦✳

❚❤❡ ❦❡② ♣r♦♣❡rt② ❚❤❡ ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡ { s i } i ≥ ✶ ❛♥♥✐❤✐❧❛t❡s ♣♦✇❡rs✿ ▲❡♠♠❛ ✭Pr♦✉❡t ✭✶✽✺✶✮✮ ▲❡t k ≥ ✵ ✱ b � = ✵ ✱ ❛♥❞ ❧❡t f ( x ) ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ d ✳ ■❢ d ≥ k ✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ F ( x ) ♦❢ ❞❡❣r❡❡ d − k s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t② ❤♦❧❞s ❢♦r ❛❧❧ x ✵ ✿ ✷ k � s i f ( x ✵ + ib ) = F ( x ✵ ) . i = ✶ ❖t❤❡r✇✐s❡✱ ✐❢ d < k ✱ t❤❡ ❛❜♦✈❡ s✉♠ ✐s ③❡r♦✳ � � s i ◮ ❙❡t u := ✹ / n ✷ ❛♥❞ ✇r✐t❡ Φ := � n − ✶ ✶ − iu i = ✶ ✶ − ( i − ✶ ) u ◮ ❚❛❦❡ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ Φ ❛♥❞ ❡①♣r❡ss ✐t ❛s ❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ❛r♦✉♥❞ ✶ / n ◮ ❯s❡ t❤❡ ❧❡♠♠❛ t♦ ♠❛❦❡ t❤❡ ❛r❡❛s a i ✬s ❜❡ ❝❧♦s❡ t♦ ✶ / n t♦ ❛ ✏❤✐❣❤ ❞❡❣r❡❡✑

❖♣❡♥ ◗✉❡st✐♦♥ ◮ ❈❛♥ ❛ ❢❛♠✐❧② ♦❢ tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧❧② ❞❡❝r❡❛s✐♥❣ ❞✐s❝r❡♣❛♥❝② ❜❡ ❝♦♥str✉❝t❡❞❄ ◮ ❚❤❛t ✐s✱ ✐s t❤❡ s♠❛❧❧❡st ❞✐s❝r❡♣❛♥❝② r❡❛❧❧② ❡①♣♦♥❡♥t✐❛❧❄