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❈✉❜✐❝❛❧✱ ❈❛t❡❣♦r✐❝❛❧ ❛♥❞ ❈♦♥t✐♥✉♦✉s ✖ ❈♦♠♣❛r✐♥❣ ❈♦❤♦♠♦❧♦❣✐❡s ❈♦♠✐♥❣ ❋r♦♠ k ✲●r❛♣❤s ❏✐❛♥❝❤❛♦ ❲✉ ❚❡①❛s ❆✫▼ ❯♥✐✈❡rs✐t② ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊❧✐③❛❜❡t❤ ●✐❧❧❛s♣② ✭❯♥✐✈❡rs✐t② ♦❢ ▼♦♥t❛♥❛✮ ◆♦✈❡♠❜❡r ✶✵✱ ✷✵✶✾ ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✶ ✴ ✷✶

� � � � � � � � � � � � ▼♦t✐✈❛t✐♦♥s ❛♥❞ ♦✈❡r✈✐❡✇ → ♣r♦✈✐❞❡ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡s → C ∗ (Λ) ❞✐r❡❝t❡❞ ❣r❛♣❤ Λ ✭❣r❛♣❤ C ∗ ✲❛❧❣❡❜r❛✮ ← ❤❡❧♣ ✉♥❞❡rst❛♥❞ ❞②♥❛♠✐❝s ← ✏✐♥✜♥✐t❡ ♣❛t❤ s♣❛❝❡ ∼ = ❝♦♥str✉❝t✐♦♥✑ st❛♥❞❛r❞ � C ∗ ( G Λ ) s②♠❜♦❧✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ❣r♦✉♣♦✐❞ G Λ ✭❣r♦✉♣♦✐❞ C ∗ ✲❛❧❣❡❜r❛✮ . ❆ ❣r❛♣❤ ✐s ❛ ✏✶✲❞✐♠❡♥s✐♦♥❛❧ ♦❜❥❡❝t✑✳ ❍✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ t✇✐st❡❞ k ✲❣r❛♣❤ C ∗ ✲❛❧❣ → ❛ ❧♦t ♠♦r❡ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡s✱ ❡✳❣✳✱ ◆❈ t♦r✐ → C ∗ (Λ , ω ) k ✲❣r❛♣❤ Λ , ✷✲❝♦❝②❝❧❡ ω ✭❆❧❧ ❯❈❚ ❑✐r❝❤❜❡r❣ ❛❧❣❡❜r❛s ❂ ❞✐r❡❝t ❧✐♠✐ts ♦❢ t❤❡s❡✮ ✏✐♥✜♥✐t❡ ♣❛t❤ s♣❛❝❡ ∼ = ❝♦♥str✉❝t✐♦♥✑ � C ∗ ( G Λ , c ω ) ✭❞②♥❛♠✐❝❛❧ s②st❡♠✮ ❣r♦✉♣♦✐❞ G Λ , ✷✲❝♦❝②❝❧❡ c ω t✇✐st❡❞ ❣r♦✉♣♦✐❞ C ∗ ✲❛❧❣ ⇒ ❉❡✜♥❡ ❛♥❞ st✉❞② t❤❡ ❝♦❤♦♠♦❧♦❣✐❡s ♦❢ Λ ❛♥❞ G Λ ✳ ❱❛r✐♦✉s ❝♦❤♦♠♦❧♦❣✐❡s✿ ❈✉❜✐❝❛❧ ❝♦❤♦♠♦❧♦❣② ❢♦r Λ ✿ ❡❛s② t♦ ❝♦♠♣✉t❡✳ ❈❛t❡❣♦r✐❝❛❧ ❝♦❤♦♠♦❧♦❣② ❢♦r Λ ✿ ❜❡tt❡r ❢✉♥❝t♦r✐❛❧✐t②✳ ❈♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣② ❢♦r G Λ ✿ ♠♦st ✢❡①✐❜❧❡✳ ❲❡ ✐❞❡♥t✐❢② t❤❡ ✜rst t✇♦ ❛♥❞ ✐♥❞✐❝❛t❡ t❤❡ r❡❧❛t✐♦♥ t♦ t❤❡ t❤✐r❞✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✷ ✴ ✷✶

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts ✶ ❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ✭ k ✲❣r❛♣❤s✮ ✷ ❈✉❜✐❝❛❧ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ✸ ●r♦✉♣♦✐❞✱ C ∗ ✲❛❧❣❡❜r❛s ❛♥❞ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣② ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✸ ✴ ✷✶

❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ❈♦♥✈❡♥t✐♦♥✿ ✵ ∈ N ✳ ▲❡t k ∈ Z + ✳ ❉❡✜♥✐t✐♦♥ ✭❑✉♠❥✐❛♥✲P❛s❦✮ ❆ k ✲❣r❛♣❤ ✐s ❛ ❝♦✉♥t❛❜❧❡ ❝❛t❡❣♦r② Λ ✇✐t❤ ❛ ❞❡❣r❡❡ ♠❛♣ d : Λ → N k s❛t✐s❢✳ d ( µν ) = d ( µ ) + d ( ν ) ✭✇r✐t❡ Λ n := d − ✶ ( n ) ✮ ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt② ✿ ✐❢ λ ∈ Λ m + n t❤❡♥ t❤❡r❡ ❛r❡ ✉♥✐q✉❡ µ ∈ Λ m ❛♥❞ ν ∈ Λ n s✳t✳ λ = µν ✳ ❍♦✇ t♦ t❤✐♥❦ ♦❢ t❤❡♠ ❛s ❣r❛♣❤s❄ ❈♦♥str✉❝t ❛ k ✲❝♦❧♦r❡❞ ❞✐r❡❝t❡❞ ❣r❛♣❤✿ ④✈❡rt✐❝❡s⑥ ❂ Λ ✵ ✱ ④❡❞❣❡s ✇✐t❤ ❝♦❧♦r i ⑥ ❂ Λ e i ✳ ✭ e ✶ , . . . , e k ❛r❡ t❤❡ st❛♥❞❛r❞ ❣❡♥❡r❛t♦rs ♦❢ N k ✮✳ ❊❛❝❤ λ ∈ Λ e i + e j = Λ e j + e i ❞❡t❡r♠✐♥❡s ❛ ❝♦♠♠✉t✐♥❣ sq✉❛r❡ ✈✐❛ t❤❡ t✇♦ µ ′ ν ′ µ ′ λ = µ ν = ν ν ′ ❢❛❝t♦r✐③❛t✐♦♥s✿ ⋔ ⋔ ⋔ ⋔ , ❞✐❛❣r❛♠✿ λ ✳ Λ e i Λ e j Λ e j Λ e i µ ❊❛❝❤ λ ∈ Λ e i + e j + e k = Λ e j + e i + e k = . . . ❞❡t❡r♠✐♥❡s ❛ ❝♦♠♠✉t✐♥❣ ❝✉❜❡ ✈✐❛ t❤❡ s✐① ❢❛❝t♦r✐③❛t✐♦♥s✱ ❛♥❞ s♦ ♦♥✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✹ ✴ ✷✶

❉❡✜♥✐t✐♦♥ ✭❑✉♠❥✐❛♥✲P❛s❦✱ r❡♣❡❛t❡❞✮ ❆ k ✲❣r❛♣❤ ✐s ❛ ❝♦✉♥t❛❜❧❡ ❝❛t❡❣♦r② Λ ✇✐t❤ ❛ ❞❡❣r❡❡ ♠❛♣ d : Λ → N k s❛t✐s❢✳ d ( µν ) = d ( µ ) + d ( ν ) ✭✇r✐t❡ Λ n := d − ✶ ( n ) ✮ ❛♥❞ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt② ✿ ✐❢ λ ∈ Λ m + n t❤❡♥ t❤❡r❡ ❛r❡ ✉♥✐q✉❡ µ ∈ Λ m ❛♥❞ ν ∈ Λ n s✳t✳ λ = µν ✳ ❊①❛♠♣❧❡s✿ ✶✲❣r❛♣❤ ❂ ✭t❤❡ ♣❛t❤ ❝❛t❡❣♦r② ♦❢✮ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤✱ d = ❧❡♥❣t❤✳ ❚❤❡ s❡♠✐❣r♦✉♣ N k ✇✐t❤ ❛ s✐♥❣❧❡ ♦❜❥❡❝t ❛♥❞ d = ✐❞✳ ❚❤❡ ♣♦s❡t ❝❛t❡❣♦r② ( N k , ≤ ) =: Ω k ✇✐t❤ t❤❡ ❞❡❣ ♦❢ t❤❡ ✉♥✐q✉❡ ♠♦r♣❤✐s♠ ( − → n , − → m ) ❢r♦♠ − → m t♦ − → n ❞❡✜♥❡❞ t♦ ❜❡ − → m − − → n ✳ ✭❛ k ✶ ✲❣r❛♣❤✮ × ✭❛ k ✷ ✲❣r❛♣❤✮ ❂ ✭❛ ( k ✶ + k ✷ ) ✲❣r❛♣❤✮✳ ❍♦✇ ❞♦ ✇❡ ✧❞r❛✇✧ ❛ k ✲❣r❛♣❤❄ ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✺ ✴ ✷✶

❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ❛s ✏❝✉❜❡ ❝♦♠♣❧❡①❡s✑✳ ✶✳ ❈♦♥str✉❝t ❛ k ✲❝♦❧♦r❡❞ ❞✐r❡❝t❡❞ ❣r❛♣❤✿ ④✈❡rt✐❝❡s⑥ ❂ Λ ✵ ✱ ④❡❞❣❡s ✇✐t❤ ❝♦❧♦r i ⑥ ❂ Λ e i ✱ ❛s ❜❡❢♦r❡✳ ✷✳ ❋♦r n = ✵ , ✶ , . . . , k ✱ ❞❡✜♥❡ t❤❡ s❡t ♦❢ n ✲❝✉❜❡s � Λ e i ✶ + ... + e in . Q n (Λ) := ✶ ≤ e i ✶ <...< e in ≤ k ∀ i ∈ { ✶ , . . . , n } ✱ ❡❛❝❤ λ ∈ Λ e i ✶ + ... + e in ⊂ Q n ❤❛s ❛ ❢r♦♥t ❢❛❝❡ F ✵ i ( λ ) ∈ Q n − ✶ ❛♥❞ ❛ ❜❛❝❦ ❢❛❝❡ F ✶ i ( λ ) ∈ Q n − ✶ s❛t✐s❢②✐♥❣ λ = F ✵ i ( λ ) µ = ν F ✶ i ( λ ) ✇✐t❤ d ( µ ) = d ( ν ) = e i ✳ ✸✳ ❇✉✐❧❞ ❛ ❝♦♠♣❧❡① ♦♥ t❤❡ k ✲❝♦❧♦r❡❞ ❣r❛♣❤✿ ✐♥❞✉❝t✐✈❡❧② ❢♦r n = ✷ , ✸ , . . . , k ✱ ❢♦r ❡❛❝❤ λ ∈ Q n ✱ ❣❧✉❡ ❛♥ n ✲❝✉❜❡ t♦ ✐ts ❢r♦♥t ❛♥❞ ❜❛❝❦ ❢❛❝❡s✱ ✐✳❡✳ F ε i ( λ ) ❢♦r i ∈ { ✶ , . . . , n } ❛♥❞ ε ∈ { ✵ , ✶ } ✳ ✭◆♦t❡✿ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♣r♦♣❡rt② ❢♦r❝❡s t❤❡ ♣❧❛❝❡♠❡♥t ♦❢ ♠❛♥② ❝✉❜❡s✦✮ ❚❤❡r❡❢♦r❡✱ Λ � ✐ts t♦♣♦❧♦❣✐❝❛❧ r❡❛❧✐③❛t✐♦♥ | Λ | ✳ ❊①❛♠♣❧❡s✿ | N k | ∼ = T k ✳ ❍♦✇ t♦ ❣❡t t❤❡ ❑❧❡✐♥ ❜♦tt❧❡ ❛s | Λ | ❄ ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✻ ✴ ✷✶

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts ✶ ❍✐❣❤❡r✲r❛♥❦ ❣r❛♣❤s ✭ k ✲❣r❛♣❤s✮ ✷ ❈✉❜✐❝❛❧ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ✭❝♦✲✮❤♦♠♦❧♦❣② ✸ ●r♦✉♣♦✐❞✱ C ∗ ✲❛❧❣❡❜r❛s ❛♥❞ ❝♦♥t✐♥✉♦✉s ❝♦❤♦♠♦❧♦❣② ❏✐❛♥❝❤❛♦ ❲✉ ✭✇✐t❤ ●✐❧❧❛s♣② ✫ ❑✉♠❥✐❛♥✮ ❈♦❤♦♠♦❧♦❣✐❡s ❝♦♠✐♥❣ ❢r♦♠ k ✲❣r❛♣❤s ◆♦✈ ✶✵ ✷✵✶✾ ✼ ✴ ✷✶