tt t t ts t - - PowerPoint PPT Presentation

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t t t t t s t rs t

tt t t ts t - - PowerPoint PPT Presentation

Sep 27, 2022 135 likes •898 views

tt t t ts t rs t ssr t r t rr

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❙t❛t❡ ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ▼✉❧t✐♣❧❡s ♦❢ t❤❡ ❚❤✉❡✲▼♦rs❡ ❙❡t

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❏♦✐♥t ✇♦r❦ ✇✐t❤ ➱♠✐❧✐❡ ❈❤❛r❧✐❡r ❛♥❞ ❈é❧✐❛ ❈✐st❡r♥✐♥♦

❙❡♣t❡♠❜❡r ✷♥❞✱ ✷✵✶✾

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✴✸✵

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SLIDE 2

❲❤❛t ❞♦ ✇❡ ✇❛♥t t♦ ❞♦ ❄

❉❡✜♥✐t✐♦♥ ▲❡t b ∈ N≥✷✳ ❆ s✉❜s❡t X ♦❢ N ✐s b✲r❡❝♦❣♥✐③❛❜❧❡ ✐❢ r❡♣b(X) ✐s r❡❣✉❧❛r✳ ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ✇♦r❦ ✇✐t❤ ✵ r❡♣ ✳ ❚❤❡♦r❡♠ ▲❡t

✷ ❛♥❞

✳ ■❢ ✐s ✲r❡❝♦❣♥✐③❛❜❧❡✱ s♦ ✐s ✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✴✸✵

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SLIDE 3

❲❤❛t ❞♦ ✇❡ ✇❛♥t t♦ ❞♦ ❄

❉❡✜♥✐t✐♦♥ ▲❡t b ∈ N≥✷✳ ❆ s✉❜s❡t X ♦❢ N ✐s b✲r❡❝♦❣♥✐③❛❜❧❡ ✐❢ r❡♣b(X) ✐s r❡❣✉❧❛r✳ ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ✇♦r❦ ✇✐t❤ ✵∗ r❡♣b(X)✳ ❚❤❡♦r❡♠ ▲❡t

✷ ❛♥❞

✳ ■❢ ✐s ✲r❡❝♦❣♥✐③❛❜❧❡✱ s♦ ✐s ✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✴✸✵

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❲❤❛t ❞♦ ✇❡ ✇❛♥t t♦ ❞♦ ❄

❉❡✜♥✐t✐♦♥ ▲❡t b ∈ N≥✷✳ ❆ s✉❜s❡t X ♦❢ N ✐s b✲r❡❝♦❣♥✐③❛❜❧❡ ✐❢ r❡♣b(X) ✐s r❡❣✉❧❛r✳ ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ✇♦r❦ ✇✐t❤ ✵∗ r❡♣b(X)✳ ❚❤❡♦r❡♠ ▲❡t b ∈ N≥✷ ❛♥❞ m ∈ N✳ ■❢ X ⊆ N ✐s b✲r❡❝♦❣♥✐③❛❜❧❡✱ s♦ ✐s mX✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✴✸✵

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❚❤❡♦r❡♠ ❬❆❧❡①❡❡✈✱ ✷✵✵✹❪ ❚❤❡ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ✵∗ r❡♣b (m N) ✐s min

N≥✵

m

❣❝❞ (m, bN) +

N−✶

n=✵

bn ❣❝❞ (bn, m)

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✸✴✸✵

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SLIDE 6

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 7

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 8

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 9

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 10

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 11

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 12

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 13

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 14

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 15

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t r❡♣✷

✶

✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 16

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t T = {n ∈ N : | r❡♣✷(n)|✶ ∈ ✷ N} .

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 17

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t T = {n ∈ N : | r❡♣✷(n)|✶ ∈ ✷ N} . T B ✵ ✵ ✶ ✶

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

slide-18

SLIDE 18

❚❤✉❡✲▼♦rs❡

✵✶✶✵✶✵✵✶✶✵✵✶✵✶✶✵ . . . ✶✵✵✶✵✶✶✵✵✶✶✵✶✵✵✶ . . . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❚❤✉❡✲▼♦rs❡ s❡t ✐s t❤❡ s❡t T = {n ∈ N : | r❡♣✷(n)|✶ ∈ ✷ N} . T B ✵, ✸ ✵, ✸ ✶, ✷ ✶, ✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✹✴✸✵

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SLIDE 19

❉❡✜♥✐t✐♦♥ ▲❡t p, q ∈ N≥✷✳ ❲❡ s❛② t❤❛t p ❛♥❞ q ❛r❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✐♥❞❡♣❡♥❞❡♥t ✐❢ pa = qb ⇒ a = b = ✵. ❚❤❡② ❛r❡ s❛✐❞ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❞❡♣❡♥❞❡♥t ♦t❤❡r✇✐s❡✳ ❚❤❡♦r❡♠ ❬❈♦❜❤❛♠✱ ✶✾✻✾❪ ▲❡t t✇♦ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✐♥❞❡♣❡♥❞❡♥t ❜❛s❡s✳ ❆ s✉❜s❡t ♦❢ ✐s ❜♦t❤ ✲r❡❝♦❣♥✐③❛❜❧❡ ❛♥❞ ✲r❡❝♦❣♥✐③❛❜❧❡ ✐✛ ✐t ✐s ❛ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ▲❡t t✇♦ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❞❡♣❡♥❞❡♥t ❜❛s❡s✳ ❆ s✉❜s❡t ♦❢ ✐s ✲r❡❝♦❣♥✐③❛❜❧❡ ✐✛ ✐t ✐s ✲r❡❝♦❣♥✐③❛❜❧❡✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✺✴✸✵

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SLIDE 20

❉❡✜♥✐t✐♦♥ ▲❡t p, q ∈ N≥✷✳ ❲❡ s❛② t❤❛t p ❛♥❞ q ❛r❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✐♥❞❡♣❡♥❞❡♥t ✐❢ pa = qb ⇒ a = b = ✵. ❚❤❡② ❛r❡ s❛✐❞ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❞❡♣❡♥❞❡♥t ♦t❤❡r✇✐s❡✳ ❚❤❡♦r❡♠ ❬❈♦❜❤❛♠✱ ✶✾✻✾❪ ▲❡t b, b′ t✇♦ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✐♥❞❡♣❡♥❞❡♥t ❜❛s❡s✳ ❆ s✉❜s❡t ♦❢ N ✐s ❜♦t❤ b✲r❡❝♦❣♥✐③❛❜❧❡ ❛♥❞ b′✲r❡❝♦❣♥✐③❛❜❧❡ ✐✛ ✐t ✐s ❛ ✜♥✐t❡ ✉♥✐♦♥ ♦❢ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ▲❡t b, b′ t✇♦ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❞❡♣❡♥❞❡♥t ❜❛s❡s✳ ❆ s✉❜s❡t ♦❢ N ✐s b✲r❡❝♦❣♥✐③❛❜❧❡ ✐✛ ✐t ✐s b′✲r❡❝♦❣♥✐③❛❜❧❡✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✺✴✸✵

slide-21

SLIDE 21

❚❤❡ r❡s✉❧t

T = {n ∈ N : | r❡♣✷(n)|✶ ∈ ✷ N} ❚❤❡♦r❡♠ ▲❡t m ∈ N ❛♥❞ p ∈ N≥✶✳ ❚❤❡♥ t❤❡ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ✵∗ r❡♣✷p(mT ) ✐s ✷k + z p

✐❢ m = k✷z ✇✐t❤ k ♦❞❞✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✻✴✸✵

slide-22

SLIDE 22

❚❤❡ ♠❡t❤♦❞

❆✉t♦♠❛t♦♥ ▲❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ a a AT ,✷p (✵, ✵)∗ r❡♣✷p (T × N)

✷

✵ ✵ r❡♣✷

✷ ✷

✵ ✵ r❡♣✷

✷ ✷

✵ r❡♣✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✼✴✸✵

slide-23

SLIDE 23

❚❤❡ ♠❡t❤♦❞

❆✉t♦♠❛t♦♥ ▲❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ a a AT ,✷p (✵, ✵)∗ r❡♣✷p (T × N) Am,✷p (✵, ✵)∗ r❡♣✷p ({(n, mn) : n ∈ N})

✷ ✷

✵ ✵ r❡♣✷

✷ ✷

✵ r❡♣✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✼✴✸✵

slide-24

SLIDE 24

❚❤❡ ♠❡t❤♦❞

❆✉t♦♠❛t♦♥ ▲❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ a a AT ,✷p (✵, ✵)∗ r❡♣✷p (T × N) Am,✷p (✵, ✵)∗ r❡♣✷p ({(n, mn) : n ∈ N}) AT ,✷p × Am,✷p (✵, ✵)∗ r❡♣✷p ({(t, mt) : t ∈ T })

✷ ✷

✵ r❡♣✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✼✴✸✵

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SLIDE 25

❚❤❡ ♠❡t❤♦❞

❆✉t♦♠❛t♦♥ ▲❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ a a AT ,✷p (✵, ✵)∗ r❡♣✷p (T × N) Am,✷p (✵, ✵)∗ r❡♣✷p ({(n, mn) : n ∈ N}) AT ,✷p × Am,✷p (✵, ✵)∗ r❡♣✷p ({(t, mt) : t ∈ T }) π

AT ,✷p × Am,✷p

✵∗ r❡♣✷p (mT )

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✼✴✸✵

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SLIDE 26

❚❤❡ ❛✉t♦♠❛t♦♥ AT ,✷p

(✵, ✵)∗ {r❡♣✷p(t, n) : t ∈ T , n ∈ N} ❙t❛t❡s ■♥✐t✐❛❧ st❛t❡ ❋✐♥❛❧ st❛t❡s ❆❧♣❤❛❜❡t ✵ ✷ ✶ ✷ ❚r❛♥s✐t✐♦♥s

✷

✐❢ ❡❧s❡ ❋♦r ❛❧❧ ✵ ✷ ✶ ✱

✷

✐❢ ✈❛❧✷ ❡❧s❡✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✽✴✸✵

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SLIDE 27

❚❤❡ ❛✉t♦♠❛t♦♥ AT ,✷p

(✵, ✵)∗ {r❡♣✷p(t, n) : t ∈ T , n ∈ N} ❙t❛t❡s T, B ■♥✐t✐❛❧ st❛t❡ T ❋✐♥❛❧ st❛t❡s T ❆❧♣❤❛❜❡t {✵, . . . , ✷p − ✶}✷ ❚r❛♥s✐t✐♦♥s δT ,✷p (X, (a, b)) = X ✐❢ a ∈ T X ❡❧s❡. ❋♦r ❛❧❧ ✵ ✷ ✶ ✱

✷

✐❢ ✈❛❧✷ ❡❧s❡✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✽✴✸✵

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SLIDE 28

❚❤❡ ❛✉t♦♠❛t♦♥ AT ,✷p

(✵, ✵)∗ {r❡♣✷p(t, n) : t ∈ T , n ∈ N} ❙t❛t❡s T, B ■♥✐t✐❛❧ st❛t❡ T ❋✐♥❛❧ st❛t❡s T ❆❧♣❤❛❜❡t {✵, . . . , ✷p − ✶}✷ ❚r❛♥s✐t✐♦♥s δT ,✷p (X, (a, b)) = X ✐❢ a ∈ T X ❡❧s❡. ❋♦r ❛❧❧ u, v ∈ {✵, . . . , ✷p − ✶}∗✱ δT ,✷p(X, (u, v)) = X ✐❢ ✈❛❧✷p(u) ∈ T X ❡❧s❡✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✽✴✸✵

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SLIDE 29

❚❤❡ ❛✉t♦♠❛t♦♥ AT ,✹

T B (✵, ✵), (✵, ✶), (✵, ✷), (✵, ✸) (✸, ✵), (✸, ✶), (✸, ✷), (✸, ✸) (✵, ✵), (✵, ✶), (✵, ✷), (✵, ✸) (✸, ✵), (✸, ✶), (✸, ✷), (✸, ✸) (✶, ✵), (✶, ✶), (✶, ✷), (✶, ✸) (✷, ✵), (✷, ✶), (✷, ✷), (✷, ✸) (✶, ✵), (✶, ✶), (✶, ✷), (✶, ✸) (✷, ✵), (✷, ✶), (✷, ✷), (✷, ✸)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✾✴✸✵

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SLIDE 30

❚❤❡ ❛✉t♦♠❛t♦♥ Am,b

(✵, ✵)∗ {r❡♣b(n, mn) : n ∈ N} ❙t❛t❡s ✵ ✶ ■♥✐t✐❛❧ st❛t❡ ✵ ❋✐♥❛❧ st❛t❡s ✵ ❆❧♣❤❛❜❡t ✵ ✶ ✷ ❚r❛♥s✐t✐♦♥s ❋♦r ❛❧❧ ✵ ✶ ✱ ❢♦r ❛❧❧ ✵ ✶ ✱ ✈❛❧ ✈❛❧

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✵✴✸✵

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SLIDE 31

❚❤❡ ❛✉t♦♠❛t♦♥ Am,b

(✵, ✵)∗ {r❡♣b(n, mn) : n ∈ N} ❙t❛t❡s ✵, . . . , m − ✶ ■♥✐t✐❛❧ st❛t❡ ✵ ❋✐♥❛❧ st❛t❡s ✵ ❆❧♣❤❛❜❡t {✵, . . . , b − ✶}✷ ❚r❛♥s✐t✐♦♥s δm,b (i, (d, e)) = j ⇔ bi + e = md + j ❋♦r ❛❧❧ ✵ ✶ ✱ ❢♦r ❛❧❧ ✵ ✶ ✱ ✈❛❧ ✈❛❧

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✵✴✸✵

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SLIDE 32

❚❤❡ ❛✉t♦♠❛t♦♥ Am,b

(✵, ✵)∗ {r❡♣b(n, mn) : n ∈ N} ❙t❛t❡s ✵, . . . , m − ✶ ■♥✐t✐❛❧ st❛t❡ ✵ ❋✐♥❛❧ st❛t❡s ✵ ❆❧♣❤❛❜❡t {✵, . . . , b − ✶}✷ ❚r❛♥s✐t✐♦♥s δm,b (i, (d, e)) = j ⇔ bi + e = md + j ❋♦r ❛❧❧ i, j ∈ {✵, . . . , m − ✶}✱ ❢♦r ❛❧❧ u, v ∈ {✵, . . . , b − ✶}∗✱ δm,b(i, (u, v)) = j ⇔ b|(u,v)|i + ✈❛❧b(v) = m ✈❛❧b(u) + j.

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✵✴✸✵

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SLIDE 33

❚❤❡ ❛✉t♦♠❛t♦♥ A✻,✹

✵ ✶ ✷ ✸ ✹ ✺

(✵, ✵) (✶, ✸) (✶, ✵) (✷, ✸) (✷, ✵) (✸, ✸) (✵, ✶) (✵, ✷) (✵, ✸) (✵, ✵) (✵, ✶) (✶, ✷) (✶, ✶) (✶, ✷) (✶, ✸) (✷, ✵) (✷, ✶) (✷, ✷) (✷, ✶) (✸, ✷) (✸, ✸) (✸, ✵) (✸, ✶) (✸, ✷)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✶✴✸✵

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SLIDE 34

❚❤❡ ♣r♦❞✉❝t ❛✉t♦♠❛t♦♥ Am,✷p × AT ,✷p

(✵, ✵)∗ {r❡♣✷p(t, mt) : t ∈ T } ❙t❛t❡s ✵ ✶ ✵ ✶ ■♥✐t✐❛❧ st❛t❡ ✵ ❋✐♥❛❧ st❛t❡s ✵ ❆❧♣❤❛❜❡t ✵ ✷ ✶ ✷ ❚r❛♥s✐t✐♦♥s

✷

✷ ✈❛❧✷ ✈❛❧✷ ❛♥❞ ✐❢ ✈❛❧✷ ❡❧s❡ ❘❡♠❛r❦ ■❢ ❛r❡ ✜①❡❞✱ t❤❡r❡ ❡①✐st ✉♥✐q✉❡ s✉❝❤ t❤❛t ✇❡ ❤❛✈❡ ❛ tr❛♥s✐t✐♦♥ ❧❛❜❡❧❡❞ ❜② ❢r♦♠ t♦ ✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✷✴✸✵

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SLIDE 35

❚❤❡ ♣r♦❞✉❝t ❛✉t♦♠❛t♦♥ Am,✷p × AT ,✷p

(✵, ✵)∗ {r❡♣✷p(t, mt) : t ∈ T } ❙t❛t❡s (✵, T), . . . , (m − ✶, T), (✵, B), . . . , (m − ✶, B) ■♥✐t✐❛❧ st❛t❡ (✵, T) ❋✐♥❛❧ st❛t❡s (✵, T) ❆❧♣❤❛❜❡t {✵, . . . , ✷p − ✶}✷ ❚r❛♥s✐t✐♦♥s δT ,✷p ((i, X), (u, v)) = (j, Y ) ⇔ ✷p|(u,v)|i + ✈❛❧✷p(v) = m ✈❛❧✷p(u) + j ❛♥❞ Y = X ✐❢ ✈❛❧✷p(u) ∈ T X ❡❧s❡. ❘❡♠❛r❦ ■❢ ❛r❡ ✜①❡❞✱ t❤❡r❡ ❡①✐st ✉♥✐q✉❡ s✉❝❤ t❤❛t ✇❡ ❤❛✈❡ ❛ tr❛♥s✐t✐♦♥ ❧❛❜❡❧❡❞ ❜② ❢r♦♠ t♦ ✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✷✴✸✵

slide-36

SLIDE 36

❚❤❡ ♣r♦❞✉❝t ❛✉t♦♠❛t♦♥ Am,✷p × AT ,✷p

(✵, ✵)∗ {r❡♣✷p(t, mt) : t ∈ T } ❙t❛t❡s (✵, T), . . . , (m − ✶, T), (✵, B), . . . , (m − ✶, B) ■♥✐t✐❛❧ st❛t❡ (✵, T) ❋✐♥❛❧ st❛t❡s (✵, T) ❆❧♣❤❛❜❡t {✵, . . . , ✷p − ✶}✷ ❚r❛♥s✐t✐♦♥s δT ,✷p ((i, X), (u, v)) = (j, Y ) ⇔ ✷p|(u,v)|i + ✈❛❧✷p(v) = m ✈❛❧✷p(u) + j ❛♥❞ Y = X ✐❢ ✈❛❧✷p(u) ∈ T X ❡❧s❡. ❘❡♠❛r❦ ■❢ i, X, v ❛r❡ ✜①❡❞✱ t❤❡r❡ ❡①✐st ✉♥✐q✉❡ j, Y , u s✉❝❤ t❤❛t ✇❡ ❤❛✈❡ ❛ tr❛♥s✐t✐♦♥ ❧❛❜❡❧❡❞ ❜② (u, v) ❢r♦♠ (i, X) t♦ (j, Y )✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✷✴✸✵

slide-37

SLIDE 37

❚❤❡ ❛✉t♦♠❛t♦♥ A✻,✹ × AT ,✹

✵T ✶T ✷T ✸T ✹T ✺T ✵B ✶B ✷B ✸B ✹B ✺B

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✸✴✸✵

slide-38

SLIDE 38

❚❤❡ ♣r♦❥❡❝t❡❞ ❛✉t♦♠❛t♦♥ π (Am,✷p × AT ,✷p)

✵∗ r❡♣✷p (mT ) = ✵∗ {r❡♣✷p(mt) : t ∈ T } ✵ ✶ ✷ ✸ ✹ ✺ ✵ ✶ ✷ ✸ ✹ ✺ ✵ ✶ ✷ ✸

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✹✴✸✵

slide-39

SLIDE 39

❚❤❡ ♣r♦❥❡❝t❡❞ ❛✉t♦♠❛t♦♥ π (Am,✷p × AT ,✷p)

✵∗ r❡♣✷p (mT ) = ✵∗ {r❡♣✷p(mt) : t ∈ T } ✵T ✶T ✷T ✸T ✹T ✺T ✵B ✶B ✷B ✸B ✹B ✺B ✵ ✶ ✷ ✸

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✹✴✸✵

slide-40

SLIDE 40

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❛✉t♦♠❛t♦♥ π

Am,✷p × AT ,✷p

✐s ❞❡t❡r♠✐♥✐st✐❝✱ ❛❝❝❡ss✐❜❧❡✱ ❝♦❛❝❝❡ss✐❜❧❡✳ Pr♦♣♦s✐t✐♦♥ ■♥ t❤❡ ❛✉t♦♠❛t♦♥

✷ ✷

✱ t❤❡ st❛t❡s ❛♥❞ ❛r❡ ❞✐s❥♦✐♥❡❞ ❢♦r ❛❧❧ ✵ ✶ ✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✺✴✸✵

slide-41

SLIDE 41

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❛✉t♦♠❛t♦♥ π

Am,✷p × AT ,✷p

✐s ❞❡t❡r♠✐♥✐st✐❝✱ ❛❝❝❡ss✐❜❧❡✱ ❝♦❛❝❝❡ss✐❜❧❡✳ Pr♦♣♦s✐t✐♦♥ ■♥ t❤❡ ❛✉t♦♠❛t♦♥ π

Am,✷p × AT ,✷p

✱ t❤❡ st❛t❡s (i, T) ❛♥❞ (i, B) ❛r❡ ❞✐s❥♦✐♥❡❞ ❢♦r ❛❧❧ i ∈ {✵, . . . , m − ✶}✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✺✴✸✵

slide-42

SLIDE 42

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✻,✹ × AT ,✹)

✵T ✶T ✷T ✸T ✹T ✺T ✵B ✶B ✷B ✸B ✹B ✺B ✵ ✶ ✷ ✸

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✻✴✸✵

slide-43

SLIDE 43

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✻,✹ × AT ,✹)

✵T ✶T ✷T ✸T ✹T ✺T ✵B ✶B ✷B ✸B ✹B ✺B ✵B

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✻✴✸✵

slide-44

SLIDE 44

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✶✼✴✸✵

slide-45

SLIDE 45

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✵✴✸✵

slide-46

SLIDE 46

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✵✴✸✵

slide-47

SLIDE 47

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✵✴✸✵

slide-48

SLIDE 48

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✵✴✸✵

slide-49

SLIDE 49

❘❡❝❛❧❧ ❋♦r m ∈ N, p ∈ N≥✶✱ ✇❡ ✇r✐t❡ m = k✷z ✇✐t❤ z ♦❞❞✳ ❋♦r ❛❧❧ ✱ ✇❡ s❡t ✐❢ ❡❧s❡✳ ❉❡✜♥✐t✐♦♥ ❋♦r ❛❧❧ ✶ ✶ ✱ ✇❡ s❡t ✵ ✷ ✶ ✵ ✷ ✶ ❲❡ ❛❧s♦ s❡t ✵ ✵ ❛♥❞ ✵ ✵ ✷ ✶

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✶✴✸✵

slide-50

SLIDE 50

❘❡❝❛❧❧ ❋♦r m ∈ N, p ∈ N≥✶✱ ✇❡ ✇r✐t❡ m = k✷z ✇✐t❤ z ♦❞❞✳ ❋♦r ❛❧❧ n ∈ N✱ ✇❡ s❡t Tn := T ✐❢ n ∈ T B ❡❧s❡✳ ❉❡✜♥✐t✐♦♥ ❋♦r ❛❧❧ ✶ ✶ ✱ ✇❡ s❡t ✵ ✷ ✶ ✵ ✷ ✶ ❲❡ ❛❧s♦ s❡t ✵ ✵ ❛♥❞ ✵ ✵ ✷ ✶

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✶✴✸✵

slide-51

SLIDE 51

❘❡❝❛❧❧ ❋♦r m ∈ N, p ∈ N≥✶✱ ✇❡ ✇r✐t❡ m = k✷z ✇✐t❤ z ♦❞❞✳ ❋♦r ❛❧❧ n ∈ N✱ ✇❡ s❡t Tn := T ✐❢ n ∈ T B ❡❧s❡✳ ❉❡✜♥✐t✐♦♥ ❋♦r ❛❧❧ j ∈ {✶, . . . , k − ✶}✱ ✇❡ s❡t [(j, T)] := {(j + kℓ, Tℓ) : ✵ ≤ ℓ ≤ ✷z − ✶} [(j, B)] :=

j + kℓ, Tℓ
: ✵ ≤ ℓ ≤ ✷z − ✶
.

❲❡ ❛❧s♦ s❡t [(✵, T)] := {(✵, T)} ❛♥❞ [(✵, B)] :=

kℓ, Tℓ : ✵ ≤ ℓ ≤ ✷z − ✶
.

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✶✴✸✵

slide-52

SLIDE 52

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✷✴✸✵

slide-53

SLIDE 53

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✷✴✸✵

slide-54

SLIDE 54

❚❤❡ ❛✉t♦♠❛t♦♥ π (A✷✹,✹ × AT ,✹)

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✷✴✸✵

slide-55

SLIDE 55

❉❡✜♥✐t✐♦♥ ❋♦r ❛❧❧ α ∈ {✵, . . . , z − ✶}✱ ✇❡ s❡t Cα :=

k✷z−α−✶ + k✷z−αℓ, Tℓ
: ✵ ≤ ℓ ≤ ✷α − ✶
.

❋♦r ❛❧❧ β ∈

✵, . . . ,
z

p

− ✷
✱ ✇❡ s❡t

Γβ :=

α∈{βp,...,(β+✶)p−✶}

Cα. ❲❡ ❛❧s♦ s❡t Γ

z p

−✶ :=
α∈
z

p

−✶
p,...,z−✶

Cα.

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✸✴✸✵

slide-56

SLIDE 56

❲❡ ❝❛♥ ❜✉✐❧❞ ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ ✇❤✐❝❤ ✐s ❛❝❝❡ss✐❜❧❡ r❡❞✉❝❡❞✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✹✴✸✵

slide-57

SLIDE 57

❲❡ ❝❛♥ ❜✉✐❧❞ ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ ✇❤✐❝❤ ✐s ❛❝❝❡ss✐❜❧❡ r❡❞✉❝❡❞✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✹✴✸✵

slide-58

SLIDE 58

❲❡ ❝❛♥ ❜✉✐❧❞ ❛ ♥❡✇ ❛✉t♦♠❛t♦♥ ✇❤✐❝❤ ✐s ❛❝❝❡ss✐❜❧❡ r❡❞✉❝❡❞✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✹✴✸✵

slide-59

SLIDE 59

✵H ✶H ✷H ✸H ✹H ✺H ✵L ✶L ✷L ✸L ✹L ✺L ✵ ✶ ✸ ✷

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✺✴✸✵

slide-60

SLIDE 60

✵ ✶ ✷ ✸

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✺✴✸✵

slide-61

SLIDE 61

❚❤❡♦r❡♠ ▲❡t m ∈ N ❛♥❞ p ∈ N≥✶✳ ❚❤❡♥ t❤❡ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ✵∗ r❡♣✷p(mT ) ✐s ❡q✉❛❧ t♦ ✷k + z p

✐❢ m = k✷z ✇✐t❤ k ♦❞❞✳

✷ ✸

✶ ✷

✼

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✻✴✸✵

slide-62

SLIDE 62

❚❤❡♦r❡♠ ▲❡t m ∈ N ❛♥❞ p ∈ N≥✶✳ ❚❤❡♥ t❤❡ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ✵∗ r❡♣✷p(mT ) ✐s ❡q✉❛❧ t♦ ✷k + z p

✐❢ m = k✷z ✇✐t❤ k ♦❞❞✳

✷ × ✸ + ✶

✷

= ✼

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✻✴✸✵

slide-63

SLIDE 63

❈♦r♦❧❧❛r②

✐✈❡♥ ❛♥② ✷p✲r❡❝♦❣♥✐③❛❜❧❡ s❡t Y ✭✈✐❛ ❛ ✜♥✐t❡ ❛✉t♦♠❛t♦♥ A r❡❝♦✲

❣♥✐③✐♥❣ ✐t✮✱ ✐t ✐s ❞❡❝✐❞❛❜❧❡ ✇❤❡t❤❡r Y = mT ❢♦r s♦♠❡ m ∈ N✳ ❚❤❡ ❞❡❝✐s✐♦♥ ♣r♦❝❡❞✉r❡ ❝❛♥ ❜❡ r✉♥ ✐♥ t✐♠❡ O(N✷) ✇❤❡r❡ N ✐s t❤❡ ♥✉♠❜❡r ♦❢ st❛t❡s ♦❢ t❤❡ ❣✐✈❡♥ ❛✉t♦♠❛t♦♥ A ✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✼✴✸✵

slide-64

SLIDE 64

❚♦ ❣♦ ❢✉rt❤❡r

❲❤❛t ❛❜♦✉t t❤❡ ❧❛♥❣✉❛❣❡ ✵∗ r❡♣✷p (mT + r) ✇❤❡r❡ r ∈ {✵, . . . , m − ✶} ❄

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-65

SLIDE 65

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✾✴✸✵

slide-66

SLIDE 66

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-67

SLIDE 67

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-68

SLIDE 68

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-69

SLIDE 69

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-70

SLIDE 70

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-71

SLIDE 71

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-72

SLIDE 72

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-73

SLIDE 73

❚❤❡ ❛✉t♦♠❛t♦♥ π

A ✷✵

✷✹,✹ × AT ,✹

❆❞❡❧✐♥❡ ▼❛ss✉✐r

❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✽✴✸✵

slide-74

SLIDE 74

❉❡✜♥✐t✐♦♥ ❋♦r ❛❧❧ ✵ ≤ α ≤ max

z

p

, | r❡♣✷p(r)|
=: L✱

R′

α =

r

✷αp

+ ℓk✷z−αp, Xℓ
: ✵ ≤ ℓ ≤ ✷αp − ✶
✐❢ α ≤
z

p

r

✷αp

+ ℓk, Xℓ
: ✵ ≤ ℓ ≤ ✷z − ✶
❡❧s❡✳

❛♥❞ Rα = R′

α \ α−✶

i=✵

R′

i .

❋♦r ❛❧❧ ✶ ✶ ❛♥❞ ❛♥❞ ❢♦r ✵ ❛♥❞ ✱

✵

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✾✴✸✵

slide-75

SLIDE 75

❉❡✜♥✐t✐♦♥ ❋♦r ❛❧❧ ✵ ≤ α ≤ max

z

p

, | r❡♣✷p(r)|
=: L✱

R′

α =

r

✷αp

+ ℓk✷z−αp, Xℓ
: ✵ ≤ ℓ ≤ ✷αp − ✶
✐❢ α ≤
z

p

r

✷αp

+ ℓk, Xℓ
: ✵ ≤ ℓ ≤ ✷z − ✶
❡❧s❡✳

❛♥❞ Rα = R′

α \ α−✶

i=✵

R′

i .

❋♦r ❛❧❧ ✶ ≤ j ≤ k − ✶ ❛♥❞ Y ∈ {T, B} ❛♥❞ ❢♦r j = ✵ ❛♥❞ Y = X✱ SY

j

= [(j, Y )] \

L

α=✵

Rα.

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✷✾✴✸✵

slide-76

SLIDE 76

❚❤❡♦r❡♠ ▲❡t m ∈ N, r ∈ {✵, . . . , m − ✶} ❛♥❞ p ∈ N≥✶✳ ▲❡t X = T ♦r N \T ✳ ❚❤❡♥✱ t❤❡ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ✵∗ r❡♣✷p (mX + r) ✐s ❡q✉❛❧ t♦ ✷k + z p

✐❢ m = k✷z ✇✐t❤ k ♦❞❞✳

❆❞❡❧✐♥❡ ▼❛ss✉✐r ❙t❛t❡ ❈♦♠♣❧❡①✐t② ❛♥❞ ❚❤✉❡✲▼♦rs❡ ❙❡t ✸✵✴✸✵