sst t sqr t r - - PowerPoint PPT Presentation

❉✐ss❡❝t✐♥❣ t❤❡ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❛❧♠♦st ❡q✉❛❧ ❛r❡❛

❙é♠✐♥❛✐r❡ ❋r❛♥❝✐❧✐❡♥ ✕ P❛r✐s ❏❡❛♥✲P❤✐❧✐♣♣❡ ▲❛❜❜é✱ ●ü♥t❡r ❘♦t❡ ❡t ●ü♥t❡r ▼✳ ❩✐❡❣❧❡r

❯♥✐✈❡rs✐té ▲✐❜r❡ ❞❡ ❇❡r❧✐♥

✷✽ ♠❛rs ✷✵✶✾

❉✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡

▲❡t n ≥ ✷✳ ❚❛s❦✿ ❉✐ss❡❝t t❤❡ sq✉❛r❡ ✐♥t♦ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳ ❈❛s❡ ❡✈❡♥ ✷ ✹ ✻ ✳ ✳ ✳

❉✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡

▲❡t n ≥ ✷✳ ❚❛s❦✿ ❉✐ss❡❝t t❤❡ sq✉❛r❡ ✐♥t♦ n tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳ ❈❛s❡ ❡✈❡♥ ✷ ✹ ✻ ✳ ✳ ✳

❉✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡

▲❡t n ≥ ✷✳ ❚❛s❦✿ ❉✐ss❡❝t t❤❡ sq✉❛r❡ ✐♥t♦ n tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳ ❈❛s❡ n ❡✈❡♥ ✷ ✹ ✻ ✳ ✳ ✳

❉✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡

▲❡t n ≥ ✷✳ ❚❛s❦✿ ❉✐ss❡❝t t❤❡ sq✉❛r❡ ✐♥t♦ n tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳ ❈❛s❡ n ❡✈❡♥ ✷ ✹ ✹ ✻ ✳ ✳ ✳

❉✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡

▲❡t n ≥ ✷✳ ❚❛s❦✿ ❉✐ss❡❝t t❤❡ sq✉❛r❡ ✐♥t♦ n tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳ ❈❛s❡ n ❡✈❡♥ ✷ ✹ ✻ ✳ ✳ ✳

❉✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡

▲❡t n ≥ ✷✳ ❚❛s❦✿ ❉✐ss❡❝t t❤❡ sq✉❛r❡ ✐♥t♦ n tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳ ❈❛s❡ n ❡✈❡♥ ✷ ✹ ✻ ✳ ✳ ✳

❚❤❡ ♦❞❞ ❝❛s❡

n = ✸ : ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹ ✺ ◗✉❡st✐♦♥✿ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛❄

❚❤❡ ♦❞❞ ❝❛s❡

n = ✸ : ← • → ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹ ✺ ◗✉❡st✐♦♥✿ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛❄

❚❤❡ ♦❞❞ ❝❛s❡

n = ✸ : ✶✴✷ ✶✴✹ ✶✴✹

• ↑

↓ ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹ ✺ ◗✉❡st✐♦♥✿ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛❄

❚❤❡ ♦❞❞ ❝❛s❡

n = ✸ : ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹

• տ

ց ✶✴✷ ✶✴✹ ✶✴✹ ✺ ◗✉❡st✐♦♥✿ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛❄

❚❤❡ ♦❞❞ ❝❛s❡

n = ✸ : ✶✴✷ ✶✴✹ ✶✴✹ ✶✴✷ ✶✴✹ ✶✴✹

• տ

ց ✶✴✷ ✶✴✹ ✶✴✹ n = ✺ : ← • → ↑ ↓ ← • → ◗✉❡st✐♦♥✿ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛❄

❚r✐❛♥❣✉❧❛t✐♦♥ ✈s ❉✐ss❡❝t✐♦♥

❋❛❝❡✲t♦✲❢❛❝❡✿ ♥♦t ❢❛❝❡✲t♦✲❢❛❝❡✿ ❚r✐❛♥❣✉❧❛t✐♦♥ ❉✐ss❡❝t✐♦♥

❚r✐❛♥❣✉❧❛t✐♦♥ ✈s ❉✐ss❡❝t✐♦♥

❋❛❝❡✲t♦✲❢❛❝❡✿ ♥♦t ❢❛❝❡✲t♦✲❢❛❝❡✿ ❚r✐❛♥❣✉❧❛t✐♦♥ ❉✐ss❡❝t✐♦♥

▼♦♥s❦②✬s ♣r♦♦❢ ❢r♦♠ t❤❡ ❜♦♦❦

❚❤❡♦r❡♠ ✭❘✐❝❤♠❛♥✕❚❤♦♠❛s✱ ▼♦♥s❦② ✭✶✾✼✵✮✮

■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳

Pr♦♦❢✳

✶✳ ❆ s♣❡❝✐❛❧ ✸ ❝♦❧♦r✐♥❣ ♦❢ t❤❡ sq✉❛r❡✳

✶✳✶ ✉s✐♥❣ ❛ ✷✲❛❞✐❝ ✈❛❧✉❛t✐♦♥ ♦♥ ✱ ❛♥❞ ❡①t❡♥❞ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✳

✷✳ ❆ r❛✐♥❜♦✇ tr✐❛♥❣❧❡ ❝❛♥♥♦t ❤❛✈❡ ❛r❡❛ ✵ ♦r ✶ ❢♦r ♦❞❞ ✳ ✸✳ ❊✈❡r② ✜♥✐t❡ ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ r❛✐♥❜♦✇ tr✐❛♥❣❧❡s✳ ❚❤✉s ❛t ❧❡❛st ♦♥❡✦

▼♦♥s❦②✬s ♣r♦♦❢ ❢r♦♠ t❤❡ ❜♦♦❦

❚❤❡♦r❡♠ ✭❘✐❝❤♠❛♥✕❚❤♦♠❛s✱ ▼♦♥s❦② ✭✶✾✼✵✮✮

■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳

Pr♦♦❢✳

✶✳ ❆ s♣❡❝✐❛❧ ✸ ❝♦❧♦r✐♥❣ ♦❢ t❤❡ sq✉❛r❡✳

✶✳✶ ✉s✐♥❣ ❛ ✷✲❛❞✐❝ ✈❛❧✉❛t✐♦♥ ♦♥ ✱ ❛♥❞ ❡①t❡♥❞ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✳

✷✳ ❆ r❛✐♥❜♦✇ tr✐❛♥❣❧❡ ❝❛♥♥♦t ❤❛✈❡ ❛r❡❛ ✵ ♦r ✶ ❢♦r ♦❞❞ ✳ ✸✳ ❊✈❡r② ✜♥✐t❡ ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ r❛✐♥❜♦✇ tr✐❛♥❣❧❡s✳ ❚❤✉s ❛t ❧❡❛st ♦♥❡✦

▼♦♥s❦②✬s ♣r♦♦❢ ❢r♦♠ t❤❡ ❜♦♦❦

❚❤❡♦r❡♠ ✭❘✐❝❤♠❛♥✕❚❤♦♠❛s✱ ▼♦♥s❦② ✭✶✾✼✵✮✮

■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳

Pr♦♦❢✳

✶✳ ❆ s♣❡❝✐❛❧ ✸ ❝♦❧♦r✐♥❣ ♦❢ t❤❡ sq✉❛r❡✳

✶✳✶ ✉s✐♥❣ ❛ ✷✲❛❞✐❝ ✈❛❧✉❛t✐♦♥ ♦♥ Q✱ ❛♥❞ ❡①t❡♥❞ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✳

✷✳ ❆ r❛✐♥❜♦✇ tr✐❛♥❣❧❡ ❝❛♥♥♦t ❤❛✈❡ ❛r❡❛ ✵ ♦r ✶ ❢♦r ♦❞❞ ✳ ✸✳ ❊✈❡r② ✜♥✐t❡ ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ r❛✐♥❜♦✇ tr✐❛♥❣❧❡s✳ ❚❤✉s ❛t ❧❡❛st ♦♥❡✦

▼♦♥s❦②✬s ♣r♦♦❢ ❢r♦♠ t❤❡ ❜♦♦❦

❚❤❡♦r❡♠ ✭❘✐❝❤♠❛♥✕❚❤♦♠❛s✱ ▼♦♥s❦② ✭✶✾✼✵✮✮

■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳

Pr♦♦❢✳

✶✳ ❆ s♣❡❝✐❛❧ ✸ ❝♦❧♦r✐♥❣ ♦❢ t❤❡ sq✉❛r❡✳

✶✳✶ ✉s✐♥❣ ❛ ✷✲❛❞✐❝ ✈❛❧✉❛t✐♦♥ ♦♥ Q✱ ❛♥❞ ❡①t❡♥❞ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✳

✷✳ ❆ r❛✐♥❜♦✇ tr✐❛♥❣❧❡ ❝❛♥♥♦t ❤❛✈❡ ❛r❡❛ ✵ ♦r ✶/n ❢♦r ♦❞❞ n✳ ✸✳ ❊✈❡r② ✜♥✐t❡ ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ r❛✐♥❜♦✇ tr✐❛♥❣❧❡s✳ ❚❤✉s ❛t ❧❡❛st ♦♥❡✦

▼♦♥s❦②✬s ♣r♦♦❢ ❢r♦♠ t❤❡ ❜♦♦❦

❚❤❡♦r❡♠ ✭❘✐❝❤♠❛♥✕❚❤♦♠❛s✱ ▼♦♥s❦② ✭✶✾✼✵✮✮

■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❞✐ss❡❝t ❛ sq✉❛r❡ ✐♥t♦ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ♦❢ ❡q✉❛❧ ❛r❡❛✳

Pr♦♦❢✳

✶✳ ❆ s♣❡❝✐❛❧ ✸ ❝♦❧♦r✐♥❣ ♦❢ t❤❡ sq✉❛r❡✳

✶✳✶ ✉s✐♥❣ ❛ ✷✲❛❞✐❝ ✈❛❧✉❛t✐♦♥ ♦♥ Q✱ ❛♥❞ ❡①t❡♥❞ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✳

✷✳ ❆ r❛✐♥❜♦✇ tr✐❛♥❣❧❡ ❝❛♥♥♦t ❤❛✈❡ ❛r❡❛ ✵ ♦r ✶/n ❢♦r ♦❞❞ n✳ ✸✳ ❊✈❡r② ✜♥✐t❡ ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ❝♦♥t❛✐♥s ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ r❛✐♥❜♦✇ tr✐❛♥❣❧❡s✳ ❚❤✉s ❛t ❧❡❛st ♦♥❡✦

❖❦✳✳✳ ❜✉t ❤♦✇ ❝❧♦s❡ ❝❛♥ t❤❡ ❛r❡❛s ❜❡❄

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

❖❦✳✳✳ ❜✉t ❤♦✇ ❝❧♦s❡ ❝❛♥ t❤❡ ❛r❡❛s ❜❡❄

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

❖❦✳✳✳ ❜✉t ❤♦✇ ❝❧♦s❡ ❝❛♥ t❤❡ ❛r❡❛s ❜❡❄

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

■♥t✉✐t✐♦♥ ♦❢ ❧♦✇ ❞✐s❝r❡♣❛♥❝②

s❡❡♠s ♥♦t ♦♣t✐♠❛❧ s❡❡♠s t❤❡ ❜❡st ♣♦ss✐❜❧❡

■♥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✶, . . . , An

◮ ❘♦♦t✲♠❡❛♥✲sq✉❛r❡ ❡rr♦r ✭❘▼❙✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✮✿

❘▼❙(D) :=

• ✶

n

n

• i=✶
• Ai − ✶

n ✷

◮ ❘❛♥❣❡✿

❘(D) = max

i,j∈[n] |Ai − Aj|

❘ ✷ ❘▼❙ ❘

▼❡❛s✉r✐♥❣ ❛r❡❛ ❞❡✈✐❛t✐♦♥

D✿ ❞✐ss❡❝t✐♦♥ ✇✐t❤ tr✐❛♥❣❧❡ ❛r❡❛s A✶, . . . , An

◮ ❘♦♦t✲♠❡❛♥✲sq✉❛r❡ ❡rr♦r ✭❘▼❙✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✮✿

❘▼❙(D) :=

• ✶

n

n

• i=✶
• Ai − ✶

n ✷

◮ ❘❛♥❣❡✿

❘(D) = max

i,j∈[n] |Ai − Aj|

❘(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✷ c✸ c✹ b✶ i✶ i✷ ◆♦❞❡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

✶ ✷ ✸ ✹

b✶ i✶ i✷ ❙✐❞❡ ♥♦❞❡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✶ i✶ i✷ ❊❞❣❡s

❙✐♠♣❧✐❝✐❛❧ ❣r❛♣❤ ♦❢ ❛ ❞✐ss❡❝t✐♦♥

❙✐❞❡ ♥♦❞❡s − → ❧✐♥❡❛r ❝♦♥str❛✐♥ts b✶ i✶ i✷ ❙✐❞❡ ♥♦❞❡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✐♦♥

❋r❛♠❡❞ ♠❛♣s

V (Γ) := ♥♦❞❡s ♦❢ t❤❡ ❣r❛♣❤ Γ ♦❢ ❛ ✜①❡❞ ❞✐ss❡❝t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❋r❛♠❡❞ ♠❛♣✮

❆ ❢r❛♠❡❞ ♠❛♣ ✐s ❛ ♠❛♣

✷ t❤❛t s❡♥❞s t❤❡ ❝♦r♥❡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✐♦♥ ✭❋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❛♠❡❞ ♠❛♣ ✐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❡✳

❉❡✜♥✐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✐❞❡ ♥♦❞❡✳

b e c d a b e c d a b e c d a b e c d a

❍♦✇ t♦ ♠❡❛s✉r❡ ❞✐s❝r❡♣❛♥❝②❄

D = {t✶, t✷, . . . , tn} : ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s L : tr✐❛♥❣❧❡s ❢r♦♠ ❧✐♥❡❛r ❝♦♥str❛✐♥ts C = s❡t ♦❢ ❝♦r♥❡r ♥♦❞❡s ΓD

❉❡✜♥✐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✷, . . . , tn} : ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s L : tr✐❛♥❣❧❡s ❢r♦♠ ❧✐♥❡❛r ❝♦♥str❛✐♥ts C = s❡t ♦❢ ❝♦r♥❡r ♥♦❞❡s ΓD

❉❡✜♥✐t✐♦♥ ✭❆r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧✮

❚❤❡ ❛r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧ πD ∈ R [XD] ♦❢ D ✐s t❤❡ ♣♦❧②♥♦♠✐❛❧

πD(XD) =

i∈[n]

• A(ti) − ✶

n

✷ +

ℓ∈L A(ℓ)✷ + v∈C

• (xv − pv)✷ + (yv − qv)✷

.

✇❤❡r❡ (pc, qc) ❛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✷, . . . , tn} : ❞✐ss❡❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ✇✐t❤ n tr✐❛♥❣❧❡s L : tr✐❛♥❣❧❡s ❢r♦♠ ❧✐♥❡❛r ❝♦♥str❛✐♥ts C = s❡t ♦❢ ❝♦r♥❡r ♥♦❞❡s ΓD

❉❡✜♥✐t✐♦♥ ✭❆r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧✮

❚❤❡ ❛r❡❛ ❞✐✛❡r❡♥❝❡ ♣♦❧②♥♦♠✐❛❧ πD ∈ R [XD] ♦❢ D ✐s t❤❡ ♣♦❧②♥♦♠✐❛❧

πD(XD) =

i∈[n]

• A(ti) − ✶

n

✷ +

ℓ∈L A(ℓ)✷ + v∈C

• (xv − pv)✷ + (yv − qv)✷

.

✇❤❡r❡ (pc, qc) ❛r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❝♦r♥❡rs ♦❢ t❤❡ sq✉❛r❡ ❛♥❞ A(ti) ❞❡♥♦t❡s t❤❡ ❛r❡❛ ♦❢ tr✐❛♥❣❧❡ ti✱ ✐✳❡✳ ❛ ❞❡t❡r♠✐♥❛♥t ♦❢ s✐③❡ ✸ A(ti) = ✶ ✷

• ✶

✶ ✶ x✶ x✷ x✸ y✶ y✷ y✸

❊①❛♠♣❧❡

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

✶ ✷ ✵ ✷ ✷ ✷

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿

• s✶
• s✷
• s✸
• s✹
• b
• a
• c
• ti∈D
• Aφ(ti) − ✶

n

✷ = ✹✼/✶✹✹✵ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿

✶ ✷ ✵ ✷ ✷ ✷

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿

• s✶
• s✷
• s✸
• s✹
• b
• a
• c
• ti∈D
• Aφ(ti) − ✶

n

✷ = ✹✼/✶✹✹✵ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿

✶ ✷ ✵ ✷ ✷ ✷

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿

• s✶
• s✷
• s✸
• s✹
• b
• a
• c
• ti∈D
• Aφ(ti) − ✶

n

✷ = ✵ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿

✶ ✷ ✵ ✷ ✷ ✷

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿

✶ ✺ ✶ ✺ ✶ ✺

• s✶
• s✷
• s✸
• s✹
• b
• a
• c
• ti∈D
• Aφ(ti) − ✶

n

✷ = ✵ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿

✶ ✷ ✵ ✷ ✷ ✷

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿

✶ ✺ ✶ ✺

• s✶
• s✷
• s✸
• s✹
• b
• a
• c
• ti∈D
• Aφ(ti) − ✶

n

✷ = ✵ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿

✶ ✷ ✵ ✷ ✷ ✷

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❞✐ss❡❝t✐♦♥✿

• s✶
• s✷
• s✸
• s✹
• b
• a
• c
• ti∈D
• Aφ(ti) − ✶

n

✷ = ✵ ■♥ t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧✿

πD(XD) =

ti∈D

• Aφ(ti) − ✶

n

✷ +

>✵

• ℓ∈L

Aφ(ℓ)✷ +

v∈C

• (xφ(v) − pv)✷ + (yφ(v) − qv)✷

.

❍♦✇ s♠❛❧❧ ❝❛♥ πD(XD) ❜❡❄

πD(XD) ≥ ✵ ❜② ❞❡✜♥✐t✐♦♥ ✵ ❞❡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♠❛❧❧ ❝❛♥ πD(XD) ❜❡❄

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

❍♦✇ s♠❛❧❧ ❝❛♥ πD(XD) ❜❡❄

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

❍♦✇ s♠❛❧❧ ❝❛♥ πD(XD) ❜❡❄

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

❍♦✇ s♠❛❧❧ ❝❛♥ πD(XD) ❜❡❄

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

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) = minT∈Dn

• maxti,tj∈T |A(ti) − A(tj)|
• ✳

✺ ✵ ✵✷✷✺ ✼ ✵ ✵✵✸✶ ✾ ✵ ✵✵✵✶✹ ❛♥❞ ✶✶ ✹ ✷ ✶✵

✻✱ ✭✇❡❛❦❧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) = minT∈Dn

• maxti,tj∈T |A(ti) − A(tj)|
• ✳

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) = minT∈Dn

• maxti,tj∈T |A(ti) − A(tj)|
• ✳

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) = minT∈Dn

• maxti,tj∈T |A(ti) − A(tj)|
• ✳

M(✺) ≤ ✵.✵✷✷✺ 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) = minT∈Dn

• maxti,tj∈T |A(ti) − A(tj)|
• ✳

M(✺) ≤ ✵.✵✷✷✺ M(✼) ≤ ✵.✵✵✸✶ M(✾) ≤ ✵.✵✵✵✶✹ ❛♥❞ M(✶✶) ≤ ✹.✷ × ✶✵−✻✱ ✭✇❡❛❦❧②✮ s✉❣❣❡st✐♥❣ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝r❡❛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 ✶

✸ ✳

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♦♦❢ 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♦♦❢ t❡❝❤♥✐q✉❡✿ ✉s❡❞ t❤❡ t❤❡♦r② ♦❢ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✳

◆❡✇ r❡s✉❧ts ✕ ▲♦✇❡r ❜♦✉♥❞

❇❡❝❛✉s❡ ❘(D) ≥ ❘▼❙(D) ❛♥❞ n❘▼❙(D)✷ = πD(XD)✱ ✐t s✉✣❝❡s t♦ ❣❡t ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r πD(XD) t♦ ❜♦✉♥❞ ❘(D)✳ ❲❡ ❣❡t ❘(D) ≥ ✶ ✷✷O(n) ✭❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✮ 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)✷ = πD(XD)✱ ✐t s✉✣❝❡s t♦ ❣❡t ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r πD(XD) 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♥♦♠✐❛❧✳

◆❡✇ 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 ❢♦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♦✈✐❞❡ ❛ ❢❛♠✐❧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♦✈✐❞❡ ❛ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s Zn ❢♦r ❡✈❡r② ♦❞❞ n ✇✐t❤ ❘(Zn) ≤ ✶ nlog✷ n−✺ = ✶ ✷Ω(log✷ n) ✭s✉♣❡r♣♦❧②♥♦♠✐❛❧✮

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

❆♥s❛t③✿ ❣❛♣ t❤❡♦r❡♠ ✐♥ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②

❚❤❡♦r❡♠ ✭❊♠✐r✐s✕▼♦✉rr❛✐♥✕❚s✐❣❛r✐❞❛s✱ ✷✵✶✵✮

■❢ f ∈ Z[x✶, . . . , xk] ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ♦♥ t❤❡ k✲s✐♠♣❧❡①✿

• x ∈ Rk

≥✵ : k

• i=✶

xi ≤ ✶

• ,

❛♥❞ f ✐s ♦❢ ❞❡❣r❡❡ d✱ ✇✐t❤ ❝♦❡✣❝✐❡♥ts ❜♦✉♥❞❡❞ ❜② ✷τ✱ 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✶, . . . , xk] ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ♦♥ t❤❡ k✲s✐♠♣❧❡①✿

• x ∈ Rk

≥✵ : k

• i=✶

xi ≤ ✶

• ,

❛♥❞ f ✐s ♦❢ ❞❡❣r❡❡ d✱ ✇✐t❤ ❝♦❡✣❝✐❡♥ts ❜♦✉♥❞❡❞ ❜② ✷τ✱ t❤❡♥ t❤❡ ♠✐♥✐♠✉♠ mDMM ♦❢ f ♦♥ t❤❡ k✲s✐♠♣❧❡① s❛t✐s✜❡s

− log mDMM < (k✷ + k) log √ d +

• k✷ log d + k(✸ + ✸ log d + τ + d log k)

+d(log k + ✶) + log d + τ + ✷] d(d − ✶)k−✶.

mDMM ✐s t❤❡ ❉❛✈❡♥♣♦rt✕▼❛❤❧❡r✕▼✐❣♥♦tt❡ ❜♦✉♥❞✳

❆♥s❛t③✿ ❣❛♣ t❤❡♦r❡♠ ✐♥ r❡❛❧ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②

❈♦r♦❧❧❛r②

❚❤❡ ♠✐♥✐♠✉♠ M ❢♦r t❤❡ ❞✐s❝r❡♣❛♥❝② ♣♦❧②♥♦♠✐❛❧ πD(XD) 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 ❜♦✉♥❞❄

■♠♣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❤ 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❡ ❛ ❝♦♠❜✐♥❛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❡ ❛ ❝♦♠❜✐♥❛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

■♠♣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

■♠♣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

■♠♣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

❈♦♠♣✉t❛t✐♦♥❛❧ ❡✈✐❞❡♥❝❡s

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

❈♦♠♣✉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 ✼ ✈❡rt✐❝❡s πD(XD) ✵✳✵✵✵✵✶✶✹✹✸✸✷✻✽ ✵✳✵✵✵✶✽✸✸✸✵✽✾✶ ❘❛♥❣❡ ✵✳✵✵✹✵✵✽✶✵ ✵✳✵✶✷✼✽✼✾ ✽ ✈❡rt✐❝❡s πD(XD) ✵✳✵✵✵✵✼✺✸✷✾✵ ✹.✷✸✺✻✻✽✾✽ × ✶✵−✻ ❘❛♥❣❡ ✵✳✵✶✵✷✶✹✾ ✵✳✵✵✷✸✷✵✻✽ n ❘▼❙ ✸ ✶.✶✼✽✺✶ × ✶✵−✶ ✺ ✶.✵✷✾✺ × ✶✵−✷ ✼ ✼.✼✼✽✼✽✽ × ✶✵−✹ ✾∗ ✷.✼✸✻✽✸✾ × ✶✵−✹

❆ ♥✐❝❡ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮

❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ❛ r❛♥❣❡ ♦r❞❡r ♦❢ ✶

✺ ✳

❆ ♥✐❝❡ ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s

❚❤❡♦r❡♠ ✭▲✳✲❘♦t❡✲❩✐❡❣❧❡r✮

❚❤✐s ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s ❤❛s ❛ r❛♥❣❡ ♦r❞❡r ♦❢ O(✶/n✺)✳

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

✶

✷

✳

❆♥ ❡✈❡♥ ♥✐❝❡r ❢❛♠✐❧② ♦❢ ❞✐ss❡❝t✐♦♥s

❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡✿ +, −, −, +, −, +, +, −, ❡t❝✳

✶/n + + + + + + + + − − − − − − − −

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

✶ nΩ(log✷ n) ✳

❊st✐♠❛t✐♥❣ t❤❡ ❡rr♦r

1/n a1 a2 a3 P Q R S O ai+1 ai . . . F G . . . E T an−1 2/n

❙❡t Ai := a✶ + · · · + ai✱ ✇❡ ❤❛✈❡ △EGO △EFO = n/✹ − Ai+✶ n/✹ − Ai ❛♥❞ RO/SO = QO/PO

❊st✐♠❛t✐♥❣ t❤❡ ❡rr♦r

1/n a1 a2 a3 P Q R S O ai+1 ai . . . F G . . . E T an−1 2/n

❚♦ ❡♥❞ ✇✐t❤ ❛ ✈❡rt✐❝❛❧ s❡❣♠❡♥t✱ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ r❛t✐♦s ♦❢ ✧✰✧ ❛♥❞ ✧✲✧ s❤♦✉❧❞ ❡q✉❛❧ RO/SO ❛♥❞ QO/PO✿

n−✶

• i=✶

n/✹ − Ai+✶ n/✹ − Ai τi

!

= ✶.

❚❤❡ ❦❡② ♣r♦♣❡rt②

❚❤❡ ❚❤✉❡✕▼♦rs❡ s❡q✉❡♥❝❡ {si}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

• i=✶

sif (x✵ + ib) = F(x✵). ❖t❤❡r✇✐s❡✱ ✐❢ d < k✱ t❤❡ ❛❜♦✈❡ s✉♠ ✐s ③❡r♦✳ ❙❡t ✹

✷ ❛♥❞ ✇r✐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✉❡♥❝❡ {si}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

• i=✶

sif (x✵ + ib) = F(x✵). ❖t❤❡r✇✐s❡✱ ✐❢ d < k✱ t❤❡ ❛❜♦✈❡ s✉♠ ✐s ③❡r♦✳

◮ ❙❡t u := ✹/n✷ ❛♥❞ ✇r✐t❡ Φ := n−✶ i=✶

• ✶−iu

✶−(i−✶)u

si

◮ ❚❛❦❡ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ Φ ❛♥❞ ❡①♣r❡ss ✐t ❛s ❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥

❛r♦✉♥❞ ✶/n

◮ ❯s❡ t❤❡ ❧❡♠♠❛ t♦ ♠❛❦❡ t❤❡ ❛r❡❛s ai✬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✐❛❧❄

▼❡r❝✐✦

❆ s♠❛❧❧ ❞✐s❝r❡♣❛♥❝② tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ✶✶ tr✐❛♥❣❧❡s