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❊①❛♠♣❧❡✿ ❆r❡❛✲✇❡✐❣❤t❡❞ ✐♥t❡❣❡r ♣❛rt✐t✐♦♥s ✐♥ ❛ N × M ❜♦①✱ ✇❤✐❝❤ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ( N + ✶ ) ✲t✉♣❧❡s ( x ✵ , x ✶ , . . . , x N ) ✱ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✈❡rt✐❝❛❧ ✐♥❝r❡♠❡♥ts ❛❧♦♥❣ t❤❡ ✈❛r✐♦✉s ❝♦❧✉♠♥s✿ � k kx k δ M , � µ N , M ( x ) ∝ q k x k ❚❤❡ ♥❛t✉r❛❧ ♣r♦❜❧❡♠✱ ✐♥ ✇❤✐❝❤ t❤❡ δ ✲❝♦♥str❛✐♥t ✐s tr❛❞❡❞ ❢♦r ❛ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ω ✱ ❤❛s ♠❡❛s✉r❡ � � k kx k ω k x k µ N , [ α ] ( x ) ∝ q ❛♥❞ t❤❡ ❜❡st ②♦✉ ❝❛♥ ❞♦ ✐s t✉♥❡ ω ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐s❡ α N , M ✳ ■♥ t❤✐s ❝❛s❡ α N , M = A ( ω ) − ✶ [ z M ] A ( ω z ) N ✶ � ✇✐t❤ A ( ω ) = ✶ − ω q k k = ✵ x = ✸ ✹ ✷ ✵ ✶ ✵ ✵ ✷ ✶ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

◆❛ï✈❡ ❝♦♠♣❧❡①✐t②✿ ❇r✐❞❣❡ ❝❛s❡ ❙❛② t❤❛t ②♦✉ ❛r❡ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡✱ ❛♥❞ t❤❛t s❛♠♣❧✐♥❣ ❢r♦♠ µ N , [ α ] t❛❦❡s t✐♠❡ T N ∼ τ N ✳ ❚❤❡♥ ♦❢ ❝♦✉rs❡ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ♦❢ ♦r❞❡r T ∼ τ N /α N , M ✳ ❊✈❡♥ ✐♥ t❤❡ ✏❜❡st r❡❛❧✐st✐❝ ❝❛s❡✑ ♦❢ ●❛✉ss✐❛♥ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ m ( C ) ∈ Z d ❛r♦✉♥❞ M ✇✐t❤ ✈❛r✐❛♥❝❡ ✭t❡♥s♦r s♣❡❝tr✉♠✮ ❧✐♥❡❛r ✐♥ N ✱ t❤✐s ❣✐✈❡s T ∼ N ✶ + d ✷ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✐♥✈❡♥t ❛ ❜❡tt❡r ❛❧❣♦r✐t❤♠ d ✷ ❛s ♠✉❝❤ ❛s ✇❡ ❝❛♥✳ ✐♥ ♦r❞❡r t♦ ❦✐❧❧ t❤❡ ❡①tr❛ ❢❛❝t♦r N ❚❤✐s ✇✐❧❧ ❜❡ ❞♦♥❡✱ ✐♥ ❛ r❡str✐❝t❡❞ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s✱ t❤r♦✉❣❤ t❤❡ tr✐❝❦ ♦❢ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✇❤✐❝❤ ✐s ✐♥ ♠② ✏●❆❙❈♦♠✷✵✶✽✑ ♣❛♣❡r✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

◆❛ï✈❡ ❝♦♠♣❧❡①✐t②✿ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❙❛② t❤❛t ②♦✉ ❛r❡ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡✱ t❤❛t s❛♠♣❧✐♥❣ ❢r♦♠ µ [ α ] t❛❦❡s t✐♠❡ T n ∼ τ n ✇❤❡♥ t❤❡ ♦✉t❝♦♠❡ | C | ✐s n ✱ ❛♥❞ T + ∼ τ N ✇❤❡♥ t❤❡ s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❛❜♦rts ✇✐t❤ ❛ ❝❡rt✐✜❝❛t✐♦♥ t❤❛t | C | ✇✐❧❧ ❜❡ ❧❛r❣❡r t❤❛♥ N ✳ ✭t❤✐s ✐s ❛❝❤✐❡✈❡❞ t❤r♦✉❣❤ ❛♥t✐❝✐♣❛t❡❞ r❡❥❡❝t✐♦♥✮ ❚❤❡♥ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ♦❢ ♦r❞❡r T ∼ τ N � α n min( n , N ) α N n ❊✈❡♥ ✐♥ t❤❡ ✏❜❡st r❡❛❧✐st✐❝ ❝❛s❡✑ ♦❢ α n ∼ z n n γ ✱ ✇✐t❤ γ > − ✶ ❛♥❞ z < ✶ t❤❛t ❝❛♥ ❜❡ t✉♥❡❞ ❛t ②♦✉r ✇✐❧❧✱ t❤✐s ❣✐✈❡s T ∼ N ✷ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✐♥✈❡♥t ❛ ❜❡tt❡r ❛❧❣♦r✐t❤♠ ✐♥ ♦r❞❡r t♦ ❦✐❧❧ t❤❡ ❡①tr❛ ❢❛❝t♦r N ❛s ♠✉❝❤ ❛s ✇❡ ❝❛♥✳ ✸ ✷ ✱ ❚❤✐s ✇✐❧❧ ✏❛❧♠♦st✑ ❜❡ ❞♦♥❡✱ ❜✉t ♦♥❧② ✉♣ t♦ r❡❛❝❤ ❝♦♠♣❧❡①✐t② N t❤r♦✉❣❤ t❤❡ tr✐❝❦ ♦❢ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✱ ✇❤✐❝❤ ✐s ✐♥ ♠② ✏●❆❙❈♦♠✷✵✶✻✑ ♣❛♣❡r✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P❧❛♥ ♦❢ t❤❡ t❛❧❦ ■t✬s ♥♦✇ ❝❧❡❛r✿ ■✬♠ tr②✐♥❣ t♦ ❵s❡❧❧✬ ②♦✉ t✇♦ r❡❝❡♥t t♦♦❧s ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ✏❇♦❧t③♠❛♥♥✕❧✐❦❡✑ ❛❧❣♦r✐t❤♠s✱ t❤❛t ✐s✱ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠s ✇❤✐❝❤ ✇♦✉❧❞ ❜❡ ❧✐♥❡❛r ✐❢ ✐t ✇❡r❡♥✬t ❢♦r s♦♠❡ s✐③❡ ❝♦♥str❛✐♥t✱ ❛♥❞ ❛r❡ ✐♥st❡❛❞ T ∼ N ✶ + γ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♠❛♥② r❡♣❡t✐t✐♦♥s ♥❡❝❡ss❛r② t♦ ❣❡t t❤❡ ❞❡s✐r❡❞ s✐③❡✳ ■ ♣r♦♣♦s❡ ②♦✉ t✇♦ ♠❛✐♥ t♦♦❧s✿ ❇♦❧t③♠❛♥♥ ❝❛s❡✿ ✉s❡ t❤❡ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦ ✐♥ ✇✐❞❡ ❣❡♥❡r❛❧✐t②✱ ❜✉t t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ♦♥❧② ❞❡❝r❡❛s❡s t♦ γ ✷ ❉✐s❝❧❛✐♠❡r✱ t❤✐s ✐s ✐♥ ♣❛rt ✏✉♥❞❡r ❝♦♥str✉❝t✐♦♥✑✦ ❇r✐❞❣❡ ❝❛s❡✿ ✐♥ s✉✐t❛❜❧❡ ❝✐r❝✉♠st❛♥❝❡s✱ ②♦✉ ❝❛♥ ✉s❡ t❤❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr✐❝❦ ❛♥❞ r❡♠♦✈❡ t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ❇❡❢♦r❡ t❤✐s✱ ■ ✇❛♥t t♦ ❞✐s❝✉ss t❤❡ t❤❡♦r❡t✐❝❛❧ ❧✐♠✐t t♦ ♣♦ss✐❜❧❡ ✐♠♣r♦✈❡♠❡♥ts✱ ❛♥❞ ❛ ♥✐❝❡ ❛❧❣♦r✐t❤♠✱ ❞✉❡ t♦ ❇❛❝❤❡r✱ ❇♦❞✐♥✐✱ ❍♦❧❧❡♥❞❡r ❛♥❞ ▲✉♠❜r♦s♦✱ t❤❛t ❝❛♥ r❡❛❝❤ t❤✐s ❧✐♠✐t ✐♥ ❛ s♣❡❝✐❛❧ ❝❛s❡✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞ ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t t❤❡ ♥❛ï✈❡ ❛❧❣♦r✐t❤♠ ❤❛s ❝♦♠♣❧❡①✐t② n ✶ + d ✷ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡✱ ❛♥❞ n ✷ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡✳ ❚❤✐s s❡❡♠s ❜❛❞✳ ❇✉t ❤♦✇ ❜❛❞ ❡①❛❝t❧②❄ ❍♦✇ ❣♦♦❞ ❝❛♥ ✇❡ ♣♦ss✐❜❧② ❞♦❄ ▲❡t ✉s tr② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ✐♥tr✐♥s✐❝ ♠✐♥✐♠❛❧ ❝♦♠♣❧❡①✐t② ♦❢ ❛ ♣r♦❜❧❡♠✳ ❚❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s ❞❡✜♥❡❞ ♦♥❧② ✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✱ ❛♥❞ ✇✐t❤ s♦♠❡ ❞❡❣r❡❡ ♦❢ ❛r❜✐tr❛r✐♥❡ss✳ ■♥st❡❛❞✱ ❢♦r t❤❡ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✱ t❤❛t ✐s✱ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ r❛♥❞♦♠ ❜✐ts ✉s❡❞ ❢♦r s❛♠♣❧✐♥❣ ❛♥ ♦❜❥❡❝t ♦❢ s✐③❡ N ✱ ❛❧s♦ t❤❡ ♦✈❡r❛❧❧ ❝♦♥st❛♥ts ❞♦ ♠❛tt❡r✳ ❚❤❡ ✐♥tr✐♥s✐❝ ♠✐♥✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t② ♦❢ ❛♥ ❡①❛❝t s❛♠♣❧✐♥❣ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ ❜② t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ♠❡❛s✉r❡✿ � S N = − µ N ( C ) ln µ N ( C ) C ∈C N ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

r❛♥❞♦♠ ✈❡❝t♦r ♦❢ ✐♥t❡❣❡rs ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮ ❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿ | ① | := � ① = ( x ✶ , . . . , x N ) ∈ N N , C N = { ① } i x i , � N µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿ | ① | := � − r❛♥❞♦♠ ✈❡❝t♦r ① = ( x ✶ , . . . , x N ) ∈ N N , C N = { ① } i x i , ← ♦❢ ✐♥t❡❣❡rs � N − ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ← Z ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮ ↑ ✈❛r✐❛❜❧❡s ❛r❡ ◆❖❚ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮ ❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿ | ① | := � − r❛♥❞♦♠ ✈❡❝t♦r ① = ( x ✶ , . . . , x N ) ∈ N N , C N = { ① } i x i , ← ♦❢ ✐♥t❡❣❡rs � N − ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ← Z t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ↑ ✈❛r✐❛❜❧❡s ❛r❡ ◆❖❚ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿ | ① | := � − r❛♥❞♦♠ ✈❡❝t♦r ① = ( x ✶ , . . . , x N ) ∈ N N , C N = { ① } i x i , ← ♦❢ ✐♥t❡❣❡rs � N − ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ← Z ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮ ↑ ✈❛r✐❛❜❧❡s ❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✿ ❞♦❛❜❧❡ ❜② ✉s✐♥❣ ♣❡r♠✉t❛t✐♦♥ s②♠♠❡tr② ❬▲✳ ❉❡✈r♦②❡✱ ✷✵✶✷❪ 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿ | ① | := � − r❛♥❞♦♠ ✈❡❝t♦r ① = ( x ✶ , . . . , x N ) ∈ N N , C N = { ① } i x i , ← ♦❢ ✐♥t❡❣❡rs � N − ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ← Z ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮ ↑ ✈❛r✐❛❜❧❡s ❛r❡ ◆❖❚ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ � � ❲❡ ❛ss✉♠❡ E ( � i x i ) = M ✱ s♦ t❤❛t µ N , [ α ] ( ① ) := µ N , M α N , M = f i ( x i ) M i 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❙♦✱ ✇❤❛t✬s t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ ❛ ♠❡❛s✉r❡ � N µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ❄ Z S [ µ N , [ α ] ] = � ❙✐♠♣❧❡ ❢❛❝t ✶✿ i S [ f i ] = Θ( N ) ❙✐♠♣❧❡ ❢❛❝t ✷✿ � � S [ µ N , [ α ] ] = − α N , m µ N , m ( ① ) ln( α N , m µ N , m ( ① )) m ① : | ① | = m � � = E α N S ( µ N , m ) + S ( α N ) ◆♦t❡ t❤❛t✱ ❜② ❈▲❚✱ S ( α N ) = Θ(ln N ) ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❙♦✱ ✇❤❛t✬s t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ ❛ ♠❡❛s✉r❡ � N µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ❄ Z S [ µ N , [ α ] ] = � ❙✐♠♣❧❡ ❢❛❝t ✶✿ i S [ f i ] = Θ( N ) ❙✐♠♣❧❡ ❢❛❝t ✷✿ � � S [ µ N , [ α ] ] = − α N , m µ N , m ( ① ) ln( α N , m µ N , m ( ① )) m ① : | ① | = m � � = E α N S ( µ N , m ) + S ( α N ) ◆♦t❡ t❤❛t✱ ❜② ❈▲❚✱ S ( α N ) = Θ(ln N ) ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❙♦✱ ✇❤❛t✬s t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ ❛ ♠❡❛s✉r❡ � N µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M ❄ Z S [ µ N , [ α ] ] = � ❙✐♠♣❧❡ ❢❛❝t ✶✿ i S [ f i ] = Θ( N ) ❙✐♠♣❧❡ ❢❛❝t ✷✿ � � S [ µ N , [ α ] ] = − α N , m µ N , m ( ① ) ln( α N , m µ N , m ( ① )) m ① : | ① | = m � � = E α N S ( µ N , m ) + S ( α N ) ◆♦t❡ t❤❛t✱ ❜② ❈▲❚✱ S ( α N ) = Θ(ln N ) ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❇② ❈▲❚✱ ❛♥❞ ❝♦♥s✐❞❡r✐♥❣ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❧✐st ❆ ❜✐t ♠♦r❡ s✉❜t❧❡✿ ( f ✶ , . . . , f N ) ✐♥t♦ t✇♦ ❧✐sts ♦❢ s✐③❡ N ✶ ❛♥❞ N ✷ = N − N ✶ ✱ ✇❡ ❤❛✈❡ � α N , m S [ µ N , m ] = − α N ✶ , m ✶ α N − N ✶ , m − m ✶ m ✶ � � � × f N ✶ , m ✶ ( ① ✶ ) f N − N ✶ , m − m ✶ ( ① ✷ ) ln f N ✶ , m ✶ ( ① ✶ ) f N − N ✶ , m − m ✶ ( ① ✷ ) ① ✶ : | ① ✶ | = m ✶ ① ✷ : | ① ✷ | = m − m ✶ � � � = α N ✶ , m ✶ α N − N ✶ , m − m ✶ S [ µ N ✶ , m ✶ ] + S [ µ N − N ✶ , m − m ✶ ] m ✶ s♦ t❤❛t t❤❡ ❛♥s❛t③ S [ f N , M + δ ] − S [ f N , M ] = O ( δ ✷ / N ) ❝❛♥ ❜❡ ♣r♦✈❡♥ s❡❧❢✲❝♦♥s✐st❡♥t✳ ❚❤❡s❡ r❡❛s♦♥✐♥❣s ✐♠♣❧② � � � � � ln N �� S [ f N , M ] = S [ f i ] ✶ + O N i ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❇② ❈▲❚✱ ❛♥❞ ❝♦♥s✐❞❡r✐♥❣ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❧✐st ❆ ❜✐t ♠♦r❡ s✉❜t❧❡✿ ( f ✶ , . . . , f N ) ✐♥t♦ t✇♦ ❧✐sts ♦❢ s✐③❡ N ✶ ❛♥❞ N ✷ = N − N ✶ ✱ ✇❡ ❤❛✈❡ � α N , m S [ µ N , m ] = − α N ✶ , m ✶ α N − N ✶ , m − m ✶ m ✶ � � � × f N ✶ , m ✶ ( ① ✶ ) f N − N ✶ , m − m ✶ ( ① ✷ ) ln f N ✶ , m ✶ ( ① ✶ ) f N − N ✶ , m − m ✶ ( ① ✷ ) ① ✶ : | ① ✶ | = m ✶ ① ✷ : | ① ✷ | = m − m ✶ � � � = α N ✶ , m ✶ α N − N ✶ , m − m ✶ S [ µ N ✶ , m ✶ ] + S [ µ N − N ✶ , m − m ✶ ] m ✶ s♦ t❤❛t t❤❡ ❛♥s❛t③ S [ f N , M + δ ] − S [ f N , M ] = O ( δ ✷ / N ) ❝❛♥ ❜❡ ♣r♦✈❡♥ s❡❧❢✲❝♦♥s✐st❡♥t✳ ❚❤❡s❡ r❡❛s♦♥✐♥❣s ✐♠♣❧② � � � � � ln N �� S [ f N , M ] = S [ f i ] ✶ + O N i ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ■♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡✱ t❤❡ ♠❡❛s✉r❡ µ N ,α ✐s t❤❡ ♦♥❡ ✐♥❞✉❝❡❞ ❜② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ A ✶ ( z ) ✱ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ❵❝♦♠❜✐♥❛t♦r✐❛❧ s♣❡❝✐✜❝❛t✐♦♥✬ s②st❡♠  A ✶ = F ✶ ( A ✶ , . . . , A k , z )   ✳ ✳ ✳   A k = F k ( A ✶ , . . . , A k , z ) ❋r♦♠ t❤❡ ♣♦s✐t✐✈✐t② ♦❢ t❤❡ F i ✬s ❝♦❡✣❝✐❡♥ts✱ ✇❡ ❝❛♥ ✐♥t❡r♣r❡t A ✶ ( z ) ❛s ❛ s✉♠ ♦✈❡r s✉✐t❛❜❧❡ ✭❝♦❧♦✉r❡❞✱ ✇❡✐❣❤t❡❞✮ tr❡❡s✱ ✇❤❡r❡ t❤❡ ❜r❛♥❝❤✐♥❣ r✉❧❡s ❛r❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ s♣❡❝✐✜❝❛t✐♦♥✱ ❛♥❞ ❞❡♣❡♥❞ ♦♥ z ✳ ❊❛❝❤ tr❡❡ ❝♦rr❡s♣♦♥❞s t♦ ♦♥❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ❡①❛❝t❧② s❛♠♣❧❡❞ ❢r♦♠ µ N ,α ❆s ❛ r❡s✉❧t✱ ❛♥② s✉❝❤ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❜❛r❡❧② s✉❜✲❝r✐t✐❝❛❧ ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ A st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✵ A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ A st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✵ A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ Az st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✶ A A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ Az st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✶ A A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BBz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✶ A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BBz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✶ A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BAAAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✶ A A B B A A A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BAAAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✶ A A B B A A A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BAzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✷ A A B B A A A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BAzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✷ A A B B A A A ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BBBzAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✷ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BBBzAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✷ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BzzBzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✹ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BzzBzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✹ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BzzBzzz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✺ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ BzzBzzz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✺ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ zzzzBzzz st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✼ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ zzzzBzzz st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✼ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ � A = A z + B ✷ + z ❊①❛♠♣❧❡✿ ❢♦r ✇❡ ❝♦✉❧❞ ❣❡t B = A ✸ + z ✷ zzzzzzzzz st❛❝❦ s✐③❡✿ ✵ ♦❜❥✳ s✐③❡✿ ✾ A A B B A A A B B ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆t ❧✐♥❡❛r ♦r❞❡r✱ t❤❡s❡ q✉❛♥t✐t✐❡s s❛t✐s❢② ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✶ ❤❛s ❛ ✉♥✐q✉❡ ❡✐❣❡♥✈❡❝t♦r ✇✐t❤ ❛❧❧ ✵✱ ✶✱ ✶ ❛♥❞ ❡✐❣❡♥✈❛❧✉❡ ③❡r♦✱ ❛❧❧ ♦t❤❡r ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ♥❡❣❛t✐✈❡✳ ❚❤✐s ✐s t❤❡ ❡✐❣❡♥✈❛❧✉❡✴✈❡❝t♦r ❛ss♦❝✐❛t❡❞ t♦ t❤❡ st❛❝❦ s✐③❡ ❡①❝✉rs✐♦♥✱ ✇❤✐❝❤ ❤❛s ❤❡✐❣❤t ✱ s♦ ✐t ✈❛♥✐s❤❡s ❛t ❧✐♥❡❛r ♦r❞❡r✳ ❋♦r t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡✱ ✇❡ ❤❛✈❡ ✶ ❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ■♥ t❤❡ ❧✐♠✐t✱ ♦♥ ❛ s✉❝❝❡ss❢✉❧ r✉♥✱ t❤❡ st❛❝❦ s✐③❡ ♣r♦✜❧❡ ✐s ❛♥ ❡①❝✉rs✐♦♥✱ ✇❤✐❧❡ t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡ ✐s ❛ str❛✐❣❤t ❧✐♥❡✱ ❜♦t❤ ♦❢ ❧❡♥❣t❤ L ❈❛❧❧ n ✶ ( t ) , . . . , n k ( t ) , s ( t ) t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts A ✶ ✱ ✳ ✳ ✳ A k ✐♥ t❤❡ st❛❝❦✱ ❛♥❞ ♦❢ t❤❡ ♦❜❥❡❝t s✐③❡✳ √ ∼ N N L = Θ( N ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ■♥ t❤❡ ❧✐♠✐t✱ ♦♥ ❛ s✉❝❝❡ss❢✉❧ r✉♥✱ t❤❡ st❛❝❦ s✐③❡ ♣r♦✜❧❡ ✐s ❛♥ ❡①❝✉rs✐♦♥✱ ✇❤✐❧❡ t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡ ✐s ❛ str❛✐❣❤t ❧✐♥❡✱ ❜♦t❤ ♦❢ ❧❡♥❣t❤ L ❈❛❧❧ n ✶ ( t ) , . . . , n k ( t ) , s ( t ) t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts A ✶ ✱ ✳ ✳ ✳ A k ✐♥ t❤❡ st❛❝❦✱ ❛♥❞ ♦❢ t❤❡ ♦❜❥❡❝t s✐③❡✳ ❆t ❧✐♥❡❛r ♦r❞❡r✱ t❤❡s❡ q✉❛♥t✐t✐❡s s❛t✐s❢② ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ n i = t − ✶ � =: � j n j ( − δ ij + A i ∂ ˙ ∂ A i F j ) | z → z ∗ j M ij n j / t A j A j → A ∗ j M ❤❛s ❛ ✉♥✐q✉❡ ❡✐❣❡♥✈❡❝t♦r ( p ✶ , . . . , p k ) ✇✐t❤ ❛❧❧ p i > ✵✱ � i p i = ✶✱ ❛♥❞ ❡✐❣❡♥✈❛❧✉❡ ③❡r♦✱ ❛❧❧ ♦t❤❡r ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ♥❡❣❛t✐✈❡✳ ❚❤✐s ✐s t❤❡ ❡✐❣❡♥✈❛❧✉❡✴✈❡❝t♦r ❛ss♦❝✐❛t❡❞ t♦ t❤❡ st❛❝❦ s✐③❡ ❡①❝✉rs✐♦♥✱ √ ✇❤✐❝❤ ❤❛s ❤❡✐❣❤t ∼ N ✱ s♦ ✐t ✈❛♥✐s❤❡s ❛t ❧✐♥❡❛r ♦r❞❡r✳ ❋♦r t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡✱ ✇❡ ❤❛✈❡ s = t − ✶ � ≃ � j n j ( z ∂ j p j ( z ∂ ˙ ∂ z F j ) | z → z ∗ ∂ z F j ) | z → z ∗ A j A j A j → A ∗ A j → A ∗ j j ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡ ◆♦t❡ ❤♦✇ t❤❡ r❛♥❞♦♠✐s❛t✐♦♥ ♦❢ t❤❡ ❣r♦✇t❤ ♣♦s✐t✐♦♥ ❤❛s ♠❛❞❡ t❤❡ ❞②♥❛♠✐❝s ❛s②♠♣t♦t✐❝❛❧❧② ❤♦♠♦❣❡♥❡♦✉s ✐♥ t✐♠❡ ✭t❤❛t ✐s✱ ❤♦♠♦❣❡♥❡♦✉s ✉♣ t♦ O ( N − ✶ ✹ ) ✢✉❝t✉❛t✐♦♥s✱ ✇❤❡♥ t❤❡ st❛❝❦ s✐③❡ ✐s ❢❛r ❢r♦♠ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s✮ ❆s ❛ r❡s✉❧t✱ ❢r♦♠ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s♣❡❝✐✜❝❛t✐♦♥✱ ❛♥❞ t❤❡ s❡❧❡❝t✐♦♥ ♦❢ t❤❡ ✏❣♦♦❞✑ ❝r✐t✐❝❛❧ ♣♦✐♥t z ∗ ✱ ✇❡ ❝❛♥ r❡❛❞ t❤❡ ❧✐♠✐t ♣❛r❛♠❡t❡rs p i ❛♥❞ ˙ s ❆♥② ●❛❧t♦♥✕❲❛ts♦♥ ❜r❛♥❝❤✐♥❣ ♦♥ ❛ t②♣❡✲ A i ♦❜❥❡❝t ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝♦♥str✉❝t✐♦♥ ✇✐t❤ ❛♥ ✐♥tr✐♥s✐❝ ❙❤❛♥♥♦♥ ❡♥tr♦♣② S i ❏✉st ❛s ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡✱ ❤❛✈✐♥❣ ❛♥ ❡①❝✉rs✐♦♥ ✐♥st❡❛❞ ♦❢ ❛ ❣❡♥❡r✐❝ ✇❛❧❦ ♦♥❧② r❡❞✉❝❡s t❤❡ ❡♥tr♦♣② ❜② ❛ ❢❛❝t♦r ∼ (ln N ) / N √ ❚❤✐s ❣✐✈❡s L = N / ˙ s + O ( N ) ✱ ❛♥❞ � � � k ( ✶ + O ( N − ✶ ✶ ✹ )) ✱ S ≃ N i = ✶ p i S i ˙ s ✇❤✐❝❤ ✐s ♦✉r ✜♥❛❧ r❡s✉❧t ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ s✉♠♠❛r② ■♥ ❝♦♥❝❧✉s✐♦♥✱ ✐♥ ♦✉r ❝❛s❡ ♦❢ st✉❞②✱ ②♦✉ ❤❛✈❡ ❛♥ ♦♣t✐♠❛❧ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐❢ t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s ❧✐♥❡❛r✱ ❛♥❞ t❤❡ r❛♥❞♦♠ ❜✐t ❝♦♠♣❧❡①✐t② T r❛♥❞ ( N ) ✐s✱ ✉♣ t♦ ❝♦rr❡❝t✐♦♥s✱ � N ❇r✐❞❣❡ ❝❛s❡✿ µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z � � � T r❛♥❞ ( N , M ) = S [ f i ] ( ✶ + o ( ✶ )) i ❇♦❧t③♠❛♥♥ ❝❛s❡✿ { A i = F i ( A ✶ , . . . , A k , z ) } i = ✶ ,..., k � N k � � T r❛♥❞ ( N ) = p i S i ( ✶ + o ( ✶ )) s ˙ i = ✶ ■ ✇✐❧❧ ♥♦t s❤♦✇ ❛♥② ❛❧❣♦r✐t❤♠ ♦❢ ♠✐♥❡ t❤❛t r❡❛❝❤❡s ♦♣t✐♠❛❧✐t②✳ ❇✉t ■ ✇✐❧❧ s❤♦✇ ②♦✉ t❤❛t ♦♣t✐♠❛❧✐t② ❡①✐sts✦ ❇② ♣r❡s❡♥t✐♥❣ ②♦✉ t❤❡ ✏♠♦t❤❡r ♦❢ ❛❧❧ ✭❇r✐❞❣❡✲❝❛s❡ ❡①❛❝t s❛♠♣❧✐♥❣✮ ❛❧❣♦r✐t❤♠s✑ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ s✉♠♠❛r② ■♥ ❝♦♥❝❧✉s✐♦♥✱ ✐♥ ♦✉r ❝❛s❡ ♦❢ st✉❞②✱ ②♦✉ ❤❛✈❡ ❛♥ ♦♣t✐♠❛❧ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐❢ t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s ❧✐♥❡❛r✱ ❛♥❞ t❤❡ r❛♥❞♦♠ ❜✐t ❝♦♠♣❧❡①✐t② T r❛♥❞ ( N ) ✐s✱ ✉♣ t♦ ❝♦rr❡❝t✐♦♥s✱ � N ❇r✐❞❣❡ ❝❛s❡✿ µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z � � � T r❛♥❞ ( N , M ) = S [ f i ] ( ✶ + o ( ✶ )) i ❇♦❧t③♠❛♥♥ ❝❛s❡✿ { A i = F i ( A ✶ , . . . , A k , z ) } i = ✶ ,..., k � N k � � T r❛♥❞ ( N ) = p i S i ( ✶ + o ( ✶ )) s ˙ i = ✶ ■ ✇✐❧❧ ♥♦t s❤♦✇ ❛♥② ❛❧❣♦r✐t❤♠ ♦❢ ♠✐♥❡ t❤❛t r❡❛❝❤❡s ♦♣t✐♠❛❧✐t②✳ ❇✉t ■ ✇✐❧❧ s❤♦✇ ②♦✉ t❤❛t ♦♣t✐♠❛❧✐t② ❡①✐sts✦ ❇② ♣r❡s❡♥t✐♥❣ ②♦✉ t❤❡ ✏♠♦t❤❡r ♦❢ ❛❧❧ ✭❇r✐❞❣❡✲❝❛s❡ ❡①❛❝t s❛♠♣❧✐♥❣✮ ❛❧❣♦r✐t❤♠s✑ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ❤✐❞❞❡♥ ✐♥ ❛ s♠❛❧❧ ❝♦r♥❡r ♦❢ t❤❡ ♣❛♣❡r ❇❛❝❤❡r✱ ❇♦❞✐♥✐✱ ❍♦❧❧❡♥❞❡r ❛♥❞ ▲✉♠❜r♦s♦✱ ▼❡r❣❡❙❤✉✤❡✿ ❆ ❱❡r② ❋❛st✱ P❛r❛❧❧❡❧ ❘❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❆❧❣♦r✐t❤♠ ❤tt♣s✿✴✴❛r①✐✈✳♦r❣✴♣❞❢✴✶✺✵✽✳✵✸✶✻✼ ❚❤❡ ♣r♦❜❧❡♠✿ ❡①❛❝t s❛♠♣❧✐♥❣ ♦❢ str✐♥❣s ✐♥ {• , ◦} n ✇✐t❤ # {•} = k ❇❇❍▲ s♦❧✈❡s ✐t ✐♥ ❧✐♥❡❛r t✐♠❡ ❛♥❞ ♦♣t✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✳ ✭✇❤✐❝❤ ✐s T r❛♥❞ ( n ) = n ( − p ln p − ( ✶ − p ) ln( ✶ − p )) + o ( n ) ✱ ✇✐t❤ p = k n ✮ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❋✐rst ♥❛ï✈❡ ✐❞❡❛✿ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♥❛ï✈❡ ❛♣♣r♦❛❝❤✳ ❙❛♠♣❧❡ n ✈❛r✐❛❜❧❡s ① = ( x ✶ , . . . , x n ) ∈ { ✵ , ✶ } n ✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥ p ✱ r❡st❛rt ✐❢ | ① | � = k ✳ ✸ ✷ ✱ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ ∼ n ❜❡❝❛✉s❡ | ① | ✐s ❞✐str✐❜✉t❡❞ r♦✉❣❤❧② ❛s ❛ ●❛✉ss✐❛♥ ♦❢ ✈❛r✐❛♥❝❡ θ ( n ) ❛♥❞ ♠❡❛♥ k ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❙❡❝♦♥❞ ♥❛ï✈❡ ✐❞❡❛✿ ♣r♦❥❡❝t ❞♦✇♥ ❢r♦♠ ❋✐s❤❡r✕❨❛t❡s ❚❤❡ ❋✐s❤❡r✕❨❛t❡s ❛❧❣♦r✐t❤♠ s❛♠♣❧❡s ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ σ ∈ S n ✇✐t❤ ♦♣t✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✿ T r❛♥❞ ( n ) ≃ ln n ! ≃ n (ln n − ✶ ) ■t ✇♦r❦s ❜② s❛♠♣❧✐♥❣ ② ∈ { ✶ } × { ✶ , ✷ } × { ✶ , ✷ , ✸ } × · · · × { ✶ , . . . , n } ✱ ❛♥❞ ❞♦✐♥❣ ❛s ❢♦❧❧♦✇s✿ 1 2 3 4 5 6 7 8 9 1 0 ❚❤❡♥✱ ❵♣r♦❥❡❝t✐♥❣ ❞♦✇♥✬ ♠❡❛♥s 1 x i = ✶ ✐✛ σ − ✶ ( i ) ≤ k 1 1 2 ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ ∼ n ln n ✱ 3 ❜❡❝❛✉s❡✱ ❡✈❡♥ ✐❢ ❋✐s❤❡r✕❨❛t❡s ✐s 6 2 ♦♣t✐♠❛❧✱ t❤❡ ♣r♦❥❡❝t✐♦♥ t❤r♦✇s 5 ❛✇❛② ♠♦st ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ 7 6 3 7 5 1 8 1 0 9 2 4 6 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❙❡❝♦♥❞ ♥❛ï✈❡ ✐❞❡❛✿ ♣r♦❥❡❝t ❞♦✇♥ ❢r♦♠ ❋✐s❤❡r✕❨❛t❡s ❚❤❡ ❋✐s❤❡r✕❨❛t❡s ❛❧❣♦r✐t❤♠ s❛♠♣❧❡s ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ σ ∈ S n ✇✐t❤ ♦♣t✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✿ T r❛♥❞ ( n ) ≃ ln n ! ≃ n (ln n − ✶ ) ■t ✇♦r❦s ❜② s❛♠♣❧✐♥❣ ② ∈ { ✶ } × { ✶ , ✷ } × { ✶ , ✷ , ✸ } × · · · × { ✶ , . . . , n } ✱ ❛♥❞ ❞♦✐♥❣ ❛s ❢♦❧❧♦✇s✿ 1 2 3 4 5 6 7 8 9 1 0 ❚❤❡♥✱ ❵♣r♦❥❡❝t✐♥❣ ❞♦✇♥✬ ♠❡❛♥s 1 x i = ✶ ✐✛ σ − ✶ ( i ) ≤ k 1 1 2 ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ ∼ n ln n ✱ 3 ❜❡❝❛✉s❡✱ ❡✈❡♥ ✐❢ ❋✐s❤❡r✕❨❛t❡s ✐s 6 2 ♦♣t✐♠❛❧✱ t❤❡ ♣r♦❥❡❝t✐♦♥ t❤r♦✇s 5 ❛✇❛② ♠♦st ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ 7 6 3 7 5 1 8 1 0 9 2 4 6 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❚❤❡ ❣♦♦❞ ✐❞❡❛✿ ❙❛♠♣❧❡ t❤❡ n ✈❛r✐❛❜❧❡s ① = ( x ✶ , . . . , x n ) ∈ { ✵ , ✶ } n ✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥ p ✱ ♦♥❡ ❜② ♦♥❡ ✉♣ t♦ ✇❤❡♥ ②♦✉ ❤❛✈❡ k ❡♥tr✐❡s x i = ✶✱ ♦r n − k ❡♥tr✐❡s x i = ✵✳ ❚❤❡♥ ❝♦♠♣❧❡t❡ ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✇✐t❤ ✇❤❛t ✐s ♥❡❡❞❡❞✱ ❋✐♥❛❧❧②✱ ♣❡r❢♦r♠ ❋✐s❤❡r✕❨❛t❡s s❤✉✤✐♥❣s ♦♥ t❤❡s❡ ❧❛st ❛❞❞❡❞ st❡♣s✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ T r❛♥❞ ( n ) = S [ µ ] + O ( √ n ln n ) ❜❡❝❛✉s❡ t❤❡ ✜♥❛❧ s❤✉✤❡s ❛r❡ ❥✉st ❛ ❢❡✇✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❚❤❡ ❣♦♦❞ ✐❞❡❛✿ ❙❛♠♣❧❡ t❤❡ n ✈❛r✐❛❜❧❡s ① = ( x ✶ , . . . , x n ) ∈ { ✵ , ✶ } n ✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥ p ✱ ♦♥❡ ❜② ♦♥❡ ✉♣ t♦ ✇❤❡♥ ②♦✉ ❤❛✈❡ k ❡♥tr✐❡s x i = ✶✱ ♦r n − k ❡♥tr✐❡s x i = ✵✳ ❚❤❡♥ ❝♦♠♣❧❡t❡ ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✇✐t❤ ✇❤❛t ✐s ♥❡❡❞❡❞✱ ❋✐♥❛❧❧②✱ ♣❡r❢♦r♠ ❋✐s❤❡r✕❨❛t❡s s❤✉✤✐♥❣s ♦♥ t❤❡s❡ ❧❛st ❛❞❞❡❞ st❡♣s✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ T r❛♥❞ ( n ) = S [ µ ] + O ( √ n ln n ) ❜❡❝❛✉s❡ t❤❡ ✜♥❛❧ s❤✉✤❡s ❛r❡ ❥✉st ❛ ❢❡✇✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬ ❚❤❡ ❣♦♦❞ ✐❞❡❛✿ ❙❛♠♣❧❡ t❤❡ n ✈❛r✐❛❜❧❡s ① = ( x ✶ , . . . , x n ) ∈ { ✵ , ✶ } n ✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥ p ✱ ♦♥❡ ❜② ♦♥❡ ✉♣ t♦ ✇❤❡♥ ②♦✉ ❤❛✈❡ k ❡♥tr✐❡s x i = ✶✱ ♦r n − k ❡♥tr✐❡s x i = ✵✳ ❚❤❡♥ ❝♦♠♣❧❡t❡ ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✇✐t❤ ✇❤❛t ✐s ♥❡❡❞❡❞✱ ❋✐♥❛❧❧②✱ ♣❡r❢♦r♠ ❋✐s❤❡r✕❨❛t❡s s❤✉✤✐♥❣s ♦♥ t❤❡s❡ ❧❛st ❛❞❞❡❞ st❡♣s✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ T r❛♥❞ ( n ) = S [ µ ] + O ( √ n ln n ) ❜❡❝❛✉s❡ t❤❡ ✜♥❛❧ s❤✉✤❡s ❛r❡ ❥✉st ❛ ❢❡✇✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ t❤❡ ❝♦❞❡ ❆❧❣♦r✐t❤♠ ✿ ❇❇❍▲ s❤✉✤✐♥❣ ❛❧❣♦r✐t❤♠ ❜❡❣✐♥ a = k , b = n − k , i = ✵❀ r❡♣❡❛t i ✰✰ ❀ ν i ← − ❇❡r♥ β ❀ ✐❢ ν i = ✶ t❤❡♥ a ✲✲ ❡❧s❡ b ✲✲ ✉♥t✐❧ a < ✵ ♦r b < ✵ ❝♦♠♣❧❡①✐t② ∼ n ❀ ✐❢ a < ✵ t❤❡♥ ¯ ν = ✵ ❡❧s❡ ¯ ν = ✶❀ ❢♦r j ← i t♦ n ❞♦ ν j = ¯ ν ❀ h ← − ❘♥❞■♥t j ❀ ❝♦♠♣❧❡①✐t② ∼ √ n ln n ❀ s✇❛♣ ν j ❛♥❞ ν h r❡t✉r♥ ν ❡♥❞ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P❧❛♥ ♦❢ t❤❡ t❛❧❦ ■ r❡❝❛❧❧ ②♦✉ t❤❛t ■✬♠ tr②✐♥❣ t♦ ❵s❡❧❧✬ ②♦✉ t✇♦ r❡❝❡♥t t♦♦❧s ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ✏❇♦❧t③♠❛♥♥✕❧✐❦❡✑ ❛❧❣♦r✐t❤♠s✱ t❤❛t ✐s✱ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❝♦♠♣❧❡①✐t② T ∼ N ✶ + γ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♠❛♥② r❡♣❡t✐t✐♦♥s ♥❡❝❡ss❛r② t♦ ❣❡t t❤❡ ❞❡s✐r❡❞ s✐③❡✳ ■ ♣r♦♣♦s❡ ②♦✉ t✇♦ ♠❛✐♥ t♦♦❧s✿ ❇♦❧t③♠❛♥♥ ❝❛s❡✿ ✉s❡ t❤❡ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦ ✐♥ ✇✐❞❡ ❣❡♥❡r❛❧✐t②✱ ❜✉t t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ♦♥❧② ❞❡❝r❡❛s❡s t♦ γ ✷ ❇r✐❞❣❡ ❝❛s❡✿ ✐♥ s✉✐t❛❜❧❡ ❝✐r❝✉♠st❛♥❝❡s✱ ②♦✉ ❝❛♥ ✉s❡ t❤❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr✐❝❦ ❛♥❞ r❡♠♦✈❡ t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ❲❡ ❛r❡ r❡❛❞② t♦ ❣♦✦ ▲❡t ✉s st❛rt ✇✐t❤ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❛♥❞ t❤❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr✐❝❦ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ♦♥❡ s❧✐❞❡ | ① | := � ① = ( x ✶ , . . . , x N ) ∈ N N , C n = { ① } , i x i � N µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z Pr♦❜❧❡♠✿ ❆ss✉♠❡ t❤❛t s❛♠♣❧✐♥❣ ❢r♦♠ ❡❛❝❤ ❞✐str✐❜✳ f i ❝♦sts O ( ✶ ) ✳ ❋✐♥❞ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t s❛♠♣❧❡s ❢r♦♠ t❤❡ ❞✐str✐❜✉t✐♦♥ µ N , M ✐♥ ❛✈❡r❛❣❡ ❧✐♥❡❛r t✐♠❡✳ 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ♦♥❡ s❧✐❞❡ | ① | := � ① = ( x ✶ , . . . , x N ) ∈ N N , C n = { ① } , i x i � N µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z ❖✉r s♦❧✉t✐♦♥✿ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts g ( x ) ∈ { ❇❡r♥ b , P♦✐ss , ●❡♦♠ b } ✱ ❛♥❞ { q i ( s ) } ✶ ≤ i ≤ n ; s ∈ N r❡❛❧ ♣♦s✐t✐✈❡✱ s✉❝❤ t❤❛t f i ( x ) = � s q i ( s ) g ∗ s ( x ) ✳ ❚❤❡♥ ♦✉r ♥❡✇ ❛❧❣♦r✐t❤♠ ❞♦❡s ✐t✦ 0.5 0.5 0.5 0.5 0.5 0.5 1 0.8 01234 01234 01234 01234 01234 01234 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.2 q i ( s ) 01234 01234 01234 01234 01234 01234 0 1 g ( x ) 0.5 0.5 0.5 0.5 0.5 0.5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0.5 0.5 0.5 0.5 f i ( x ) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t✇♦ s❧✐❞❡s ❖✉r ♥❡✇ tr✐❝❦ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡❛s✿ ◮ ❘❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❛♥ ❡①tr❛ ❢❛❝t♦r ✐♥ t❤❡✐r ❝♦♠♣❧❡①✐t②✱ ♦♥ t❤❡ s❝❛❧❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡✳ ■♥ ♦r❞❡r t♦ ❤❛✈❡ t❤❡ ♦♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② s❝❛❧✐♥❣✱ ②♦✉ ♥❡❡❞ t❤❡ ❛✈❡r❛❣❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ t♦ ❜❡ Θ( ✶ ) ✱ ✐✳❡✳ ♥♦t t♦ s❝❛❧❡ ✇✐t❤ t❤❡ s✐③❡ n ✳ ◮ P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡s f i ( x ) = � s q i ( s ) g ∗ s ( x ) ✳ � N ❆s ❛ r❡s✉❧t t❤❡ ♠❡❛s✉r❡ µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z ✐s ❛ ♠❛r❣✐♥❛❧ ♦❢ ❛ ♠❡❛s✉r❡ ✐♥ t✇♦ s❡ts ♦❢ ✈❛r✐❛❜❧❡s✿ � N � � µ N , M ( ① , s ) = ✶ q i ( s i ) g ∗ s i ( x i ) × δ | ① | , M ✳ i = ✶ Z ◮ ❨♦✉ ❝❛♥ ✜rst s❛♠♣❧❡ s ✱ ✇✐t❤ ♠❡❛s✉r❡ µ ✶ ( s ) = � N i = ✶ q i ( s i ) ✱ t❤❡♥ ❛❝❝❡♣t t❤✐s ✈❡❝t♦r s ✇✐t❤ r❛t❡ a ( s ) ∝ g ∗| s | ( M ) ✱ ❛♥❞ ✜♥❛❧❧② s❛♠♣❧❡ ① ✇✐t❤ ♠❡❛s✉r❡ µ ✷ ( ① | s ) = � N i = ✶ g ∗ s i ( x i ) ✳ ◮ ❚❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ ✐s ❤✐❣❤ ❜❡❝❛✉s❡✱ ❛❧t❤♦✉❣❤ g ∗| s | ( M ) = Θ( N − ✶ ✷ ) ✱ ✇❡ ❤❛✈❡ g ∗| s | ( M ) / max N ( g ∗ N ( M )) = Θ( ✶ ) ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ r❡♠✐♥❞❡r ♦❢ t❤❡ ♥❛ï✈❡ ❛❧❣♦r✐t❤♠ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❯♣ t♦ r❡❞❡✜♥✐♥❣ t❤❡ f i ✬s✱ ✇❡ ❝❛♥ ❛ss✉♠❡ ✇✳❧✳♦✳❣✳ t❤❛t E ( | ① | ) = � i E [ f i ] = M ✳ ❆ss✉♠❡ t❤❛t ❜♦t❤ M ❛♥❞ σ ✷ := � i V ❛r [ f i ] ❛r❡ Θ( N ) ✳ ❚❤❡ ❵❇r✐❞❣❡ ❝❛s❡✬ r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✇♦✉❧❞ ❣✐✈❡✿ ❝♦♠♣❧❡①✐t② ∼ N ✸ / ✷ ❆❧❣♦r✐t❤♠ ✿ ◆❛ï✈❡ r❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❜❡❣✐♥ r❡♣❡❛t | ① | = ✵❀ ❢♦r i ← ✶ t♦ N ❞♦ ← ❝♦♠♣❧❡①✐t② ∼ N ❀ x i ⇐ f i | ① | ✰❂ x i ❀ √ ✉♥t✐❧ | ① | = M ← ❝♦♠♣❧❡①✐t② ∼ N ❀ r❡t✉r♥ ( x ✶ , . . . , x N ) ❡♥❞ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ r❡❥❡❝t✐♦♥ ♣❛r❛❞✐❣♠ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐♥ ❛♥② r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠✱ ②♦✉ ✇❛♥t t♦ ❞♦ ❡①❛❝t s❛♠♣❧✐♥❣ ❢♦r µ ( ① ) ✱ ✇❤❡♥ µ ( ① ) ∝ µ ✵ ( ① ) a ( ① ) ✱ ✇✐t❤ a ( ① ) ∈ [ ✵ , ✶ ] ✱ s✉♣♣♦s✐♥❣ t❤❛t ②♦✉ ❦♥♦✇ ❤♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ µ ✵ T [ µ ] ∼ T [ µ ✵ ] E ( a ( ① )) − ✶ ❆❧❣♦r✐t❤♠ ✿ ❘❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❜❡❣✐♥ r❡♣❡❛t ① ⇐ µ ✵ ; ← ❝♦♠♣❧❡①✐t② T [ µ ✵ ] α ⇐ ❇❡r♥ a ( ① ) ❀ ← ❝♦♠♣❧❡①✐t② E ( a ( ① )) − ✶ ❀ ✉♥t✐❧ α = ✶ r❡t✉r♥ ① ❡♥❞ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ r❡❥❡❝t✐♦♥ ♣❛r❛❞✐❣♠ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s ◆♦✇ ❛ss✉♠❡ µ ( ① ) ∝ � ② µ ✶ ( ② ) µ ✷ ( ① | ② ) a ( ② ) ✱ ✇✐t❤ a ( ② ) ∈ [ ✵ , ✶ ] ✱ s✉♣♣♦s✐♥❣ t❤❛t ②♦✉ ❦♥♦✇ ❤♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ µ ✶ ✱ ❛♥❞ µ ✷ ( · | ② ) ❆❧❣♦r✐t❤♠ ✿ ❘❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s ❜❡❣✐♥ r❡♣❡❛t ② ⇐ µ ✶ ; ← s❛♠♣❧❡ ❛ t❡♥t❛t✐✈❡ ② ✇✐t❤ µ ✶ α ⇐ ❇❡r♥ a ( ② ) ✉♥t✐❧ α = ✶ ← ❛❝❝❡♣t ② ✇✐t❤ r❛t❡ a ( ② ) ❀ ① ⇐ µ ✷ ( · | ② ) ; ← s❛♠♣❧❡ ① ✇✐t❤ µ ✷ ( · | ② ) r❡t✉r♥ ① ❡♥❞ � � � ② µ ✶ ( ② ) T ✶ ( ② ) + a ( ② ) T ✷ ( ② ) ≤ T ♠❛① = E ( T ✶ + aT ✷ ) E ( a ) + T ♠❛① ✶ T = � ✷ ② µ ✶ ( ② ) a ( ② ) E ( a ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ r❡❥❡❝t✐♦♥ ♣❛r❛❞✐❣♠ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s ◆♦✇ ❛ss✉♠❡ µ ( ① ) ∝ � ② µ ✶ ( ② ) µ ✷ ( ① | ② ) a ( ② ) ✱ ✇✐t❤ a ( ② ) ∈ [ ✵ , ✶ ] ✱ s✉♣♣♦s✐♥❣ t❤❛t ②♦✉ ❦♥♦✇ ❤♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ µ ✶ ✱ ❛♥❞ µ ✷ ( · | ② ) ❆❧❣♦r✐t❤♠ ✿ ❘❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s ❜❡❣✐♥ r❡♣❡❛t ② ⇐ µ ✶ ; ← ❝♦♠♣❧❡①✐t② T ✶ ( ② ) α ⇐ ❇❡r♥ a ( ② ) ← ❝♦♠♣❧❡①✐t② a ( ② ) − ✶ ❀ ✉♥t✐❧ α = ✶ ① ⇐ µ ✷ ( · | ② ) ; ← ❝♦♠♣❧❡①✐t② T ✷ ( ② ) r❡t✉r♥ ① ❡♥❞ � � � ② µ ✶ ( ② ) T ✶ ( ② ) + a ( ② ) T ✷ ( ② ) ≤ T ♠❛① = E ( T ✶ + aT ✷ ) E ( a ) + T ♠❛① ✶ T = � ✷ ② µ ✶ ( ② ) a ( ② ) E ( a ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♣r♦✈✐❞❡s ❛ ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡ P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ t❡❧❧s t❤❛t✱ ❢♦r ❛❧❧ i ✱ f i ( x ) = � s q i ( s ) g ∗ s ( x ) ✱ ✇✐t❤ q i ( s ) ≥ ✵✳ ❋r♦♠ t❤❡ ♥♦r♠❛❧✐s❛t✐♦♥ ♦❢ t❤❡ f i ✬s ❛♥❞ ♦❢ g ✱ ✐t ❢♦❧❧♦✇s t❤❛t ❛❧s♦ t❤❡ q i ( s ) ❛r❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✳ � N ❆s ❛ r❡s✉❧t t❤❡ ♠❡❛s✉r❡ µ N , M ( ① ) = ✶ i = ✶ f i ( x i ) × δ | ① | , M Z ✐s ❛ ♠❛r❣✐♥❛❧ ♦❢ ❛ ♠❡❛s✉r❡ ✐♥ t✇♦ s❡ts ♦❢ ✈❛r✐❛❜❧❡s✿ � � � N µ N , M ( ① , s ) = ✶ q i ( s i ) g ∗ s i ( x i ) × δ | ① | , M ✳ i = ✶ Z ❚❤✐s ✐s ❡①❛❝t❧② ❛s ✐♥ ❛ ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡✱ ✇✐t❤ ❝♦rr❡s♣♦♥❞❡♥❝❡ µ ✶ ( s ) = � n s❛♠♣❧❡ s ✇✐t❤ ♠❡❛s✉r❡ µ ✶ ( s ) i = ✶ q i ( s i ) a ( s ) ∝ g ∗| s | ( m ) ❛❝❝❡♣t s ✇✐t❤ r❛t❡ a ( s ) µ ✷ ( ① | s ) = � n i = ✶ g ∗ s i ( x i ) s❛♠♣❧❡ ① ✇✐t❤ ♠❡❛s✉r❡ µ ✷ ( ① | s ) ◆♦t❡✿ ❛❧t❤♦✉❣❤ µ ✶ ❛♥❞ µ ✷ ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❡❝t♦r s ✱ t❤❡ r❛t❡ a ♦♥❧② ❞❡♣❡♥❞s ♦♥ | s | = � i s i ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

■♥❝r❡❛s✐♥❣ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ ❚❤❡ ❝r✉❝✐❛❧ ♣♦✐♥t ✐s t❤❛t t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛❧❧♦✇s t♦ ✐♥❝r❡❛s❡ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡✦ ■♥ t❤❡ ❵♦r❞✐♥❛r②✬ r❡❥❡❝t✐♦♥ s❝❤❡♠❡✱ ②♦✉ ❛❝❝❡♣t ① ✐✛ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❡✈❡♥t ♦❝❝✉rs ✭✐♥ ♦✉r ❝❛s❡✱ | ① | = M ✮✳ ■❢ t❤✐s ♣r♦❜❛❜✐❧✐t② ✐s ✐♥tr✐♥s✐❝❛❧❧② s♠❛❧❧ ✭✐♥ ♦✉r ❝❛s❡✱ Θ( N − ✶ / ✷ ) ✮✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ②♦✉ ❝❛♥ ❞♦✳ ■♥ t❤❡ r❡❥❡❝t✐♦♥ s❝❤❡♠❡ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s✱ t❤❡ r❛t❡ a ( s ) ✐s ❞❡✜♥❡❞ ✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❢❛❝t♦r✱ ❛s ❧♦♥❣ ❛s max s a ( s ) ≤ ✶✳ ❍❡r❡✱ t❤❡ ♦❜✈✐♦✉s ❝❤♦✐❝❡ ❢♦r a ( s ) ✐s a ( s ) = g ∗| s | ( M ) ✱ ✇❤✐❝❤ ✐s Θ( N − ✶ ✷ ) ✳ g ∗| s | ( M ) ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♣✉s❤ ✐t ✉♣ t♦ a ( s ) = max n ( g ∗ n ( M )) ✳ ❆s ✇❡ ✇✐❧❧ s❡❡✱ ✇✐t❤ t❤✐s ❝❤♦✐❝❡ E ( a ( s )) = Θ( ✶ ) ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❍♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ ❇❡r♥ a ( s ) ❚❤✐s ✐❞❡❛ ✐s ♥♦t s✉✣❝✐❡♥t ❜② ✐ts❡❧❢✳ ❊✈❡♥ ✐❢ ②♦✉ ❦♥♦✇ ✐♥ ❛❞✈❛♥❝❡ t❤❛t✱ ❛❢t❡r ♠❛①✐♠✐s❛t✐♦♥✱ E ( a ( s )) = Θ( ✶ ) ✱ ②♦✉ st✐❧❧ ❤❛✈❡ ❛ ♣r♦❜❧❡♠✿ s❛♠♣❧✐♥❣ ❛ ❇❡r♥♦✉❧❧✐ r♥❞ ✈❛r ✇✐t❤ ♣❛r❛♠❡t❡r a ( s ) ✐s ❞✐✣❝✉❧t ✐❢ ②♦✉ ❞♦ ♥♦t ❤❛✈❡ ❛♥ ❛♥❛❧②t✐❝ ❡①♣r❡ss✐♦♥ ❢♦r a ( s ) ✳ ■t ✐s ♥♦t ❝♦♠♣✉❧s♦r② t♦ ❤❛✈❡ ❛♥ ❛♥❛❧②t✐❝ ❡①♣r❡ss✐♦♥ ❢♦r a ( s ) ✭❥✉st t❤✐♥❦ t♦ ❤♦✇ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ❛❧❣♦r✐t❤♠✿ x ⇐ ❘♥❞ [ ✵ , ✶ ] ❀ y ⇐ ❘♥❞ [ ✵ , ✶ ] ❀ r❡t✉r♥ s✐❣♥ ( ✶ − x ✷ − y ✷ ) s❛♠♣❧❡s ❇❡r♥ π/ ✹ ✇✐t❤♦✉t ❦♥♦✇✐♥❣ π ✳ ✳ ✳ ✮ ❤♦✇❡✈❡r✱ ✐t ♠❛❦❡s ❧✐❢❡ ❡❛s✐❡r✱ ❛♥❞ ✐♥ ♦✉r ❝❛s❡ ✇❡ ❤❛✈❡ ✐t ❢♦r ❢r❡❡ ✐❢ ✇❡ ❝❤♦♦s❡ t❤❡ ❜❛s❡ ❢✉♥❝t✐♦♥ g ( x ) ❢♦r ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❤❡ ❧✐st g ( x ) ∈ { ❇❡r♥ b , P♦✐ss , ●❡♦♠ b } ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❍♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ ❇❡r♥ a ( s ) ❊①❛♠♣❧❡ ✇✐t❤ ❇❡r♥♦✉❧❧✐ ✭t❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ s✐♠✐❧❛r✮ ✭❥✉st ✇r✐t❡ a ( s ) ❢♦r a ( s ) ✱ ✇✐t❤ s = | s | ✮ b M ( ✶ − b ) s − M � s � g ∗ s ( M ) M a ( s ) = max n ( g ∗ n ( M )) = b M ( ✶ − b ) n − M � n � �� max n M ❚❤❡ ♠❛① ✐s r❡❛❧✐s❡❞ ❢♦r n = ¯ n := ⌊ M / b ⌋ ✱ t❤✉s n s !(¯ n − M )! a ( s ) = ( ✶ − b ) s − ¯ n !( s − M )! ¯ ●♦♦❞ ♥❡✇s ✶✿ ❚❤✐s ✐s ❡❛s✐❧② ❡✈❛❧✉❛t❡❞ t♦ ❤✐❣❤ ♣r❡❝✐s✐♦♥ ✭✐✳❡✳✱ ❝❛❧❝✉❧❛t✐♥❣ d ❜✐♥❛r② ❞✐❣✐ts ❤❛s ❝♦♠♣❧❡①✐t② ≪ ✷ d ✮✱ s♦ t❤❛t t❤❡ ❛✈❡r❛❣❡ ❝♦st ♦❢ ❇❡r♥ a ( s ) ✐s Θ( ✶ ) ✳ ●♦♦❞ ♥❡✇s ✷✿ ❋♦r ❧❛r❣❡ M ✱ ❛♥❞ b = Θ( ✶ ) ✱ a ( s ) ❝♦♥✈❡r❣❡s t♦ ❛♥ ✉♥✲♥♦r♠❛❧✐s❡❞ ●❛✉ss✐❛♥ ❝❡♥t❡r❡❞ ❛r♦✉♥❞ ¯ n ✱ ❛♥❞ ♦❢ ✈❛r✐❛♥❝❡ Θ( M ) ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ r♦✉❣❤ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣❧❡①✐t② ❘❡❝❛❧❧ t❤❡ ❜❛s✐❝ st❡♣s ✐♥ t❤❡ r❡❥❡❝t✐♦♥ ❛❧❣♦ ❢♦r ♦✉r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡✿ µ ✶ ( s ) = � n s❛♠♣❧❡ s ✇✐t❤ ♠❡❛s✉r❡ µ ✶ ( s ) i = ✶ q i ( s i ) a ( s ) = g ∗ s ( M ) / g ∗ ¯ n ( M ) ❛❝❝❡♣t s ✇✐t❤ r❛t❡ a ( s ) µ ✷ ( ① | s ) = � n i = ✶ g ∗ s i ( x i ) s❛♠♣❧❡ ① ✇✐t❤ ♠❡❛s✉r❡ µ ✷ ( ① | s ) ❛♥❞ t❤❛t t❤✐s ❛❧❣♦r✐t❤♠ ❤❛s ❝♦♠♣❧❡①✐t② T ♠❛① E µ ✶ ( a ( s )) + T ♠❛① ✇❤❡r❡ T ♠❛① , T ♠❛① ✶ T ≤ = Θ( n ) . ✷ ✶ ✷ ❯♥❞❡r ♠✐❧❞ ❈▲❚ ❤②♣♦t❤❡s❡s✱ t❤❡ ♠❡❛s✉r❡ ♦♥ s = | s | ✐♥❞✉❝❡❞ ❜② µ ✶ ( s ) n ✱ ✇✐t❤ ✈❛r✐❛♥❝❡ σ ✷ ✐s ❛ ✭♥♦r♠❛❧✐s❡❞✮ ●❛✉ss✐❛♥ ❝❡♥t❡r❡❞ ✐♥ ¯ ✶ N ✱ n ✱ ✇✐t❤ ✈❛r✐❛♥❝❡ σ ✷ ✇❤✐❧❡ a ( s ) ✐s ❛♥ ✉♥✲♥♦r♠❛❧✐s❡❞ ●❛✉ss✐❛♥✱ ❝❡♥t❡r❡❞ ✐♥ ¯ ✷ N ✿ � � � � � − x ✷ ✶ ✶ ✶ + ✶ σ ✷ √ √ E ( a ) ≃ ✶ N exp = ❞ x ✷ N σ ✷ σ ✷ ✷ πσ ✷ σ ✷ ✶ + σ ✷ ✷ ✷ � T � T ♠❛① ✶ + ( σ ✶ /σ ✷ ) ✷ + T ♠❛① = Θ( N ) ✶ ✷ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♣r❡❝✐s❡ r❡s✉❧t ❚❤❡ t❤r❡❡ ❢✉♥❞❛♠❡♥t❛❧ ❞✐str✐❜✉t✐♦♥s β ( r ) = β r ( ✶ − β ) s − r � s �  ❇❡r♥ ∗ s β ∈ ] ✵ , ✶ [  r P♦✐ss s ( r ) = e − s s r g ∗ s β = ✵ β ( r ) = r ! − β ( r ) = | β | r ( ✶ + | β | ) − s − r � s + r − ✶ �  ●❡♦♠ ∗ s β ∈ ] − ∞ , ✵ [ r ❛r❡ s✉❝❤ t❤❛t g ∗ s α ❤❛s ❛ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ g β ✐✛ α ≤ β ✳ δ x , ✶ ●❡♦♠ − β P♦✐ss ❇✐♥♦ β s s β −∞ ✵ ✶ ❋♦r t❤❡ ❧✐st ♦❢ ❢✉♥❝t✐♦♥s F = { f ✶ , . . . , f n } ✐♥ ♦✉r ♠❡❛s✉r❡✱ ❝❛❧❧ β ♠✐♥ ( F ) t❤❡ s♠❛❧❧❡st ✈❛❧✉❡ ♦❢ β s✉❝❤ t❤❛t ❛❧❧ t❤❡ f i ✬s ❤❛✈❡ ❛ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ g β ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♣r❡❝✐s❡ r❡s✉❧t ❚❤❡♥✱ t❤❡ ❧❛r❣❡st ✈❛❧✉❡ ❢♦r E ( a ) t❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✇✐t❤✐♥ ♦✉r ❢r❛♠❡✇♦r❦ ✐s � � i E [ f i ] � a ♠❛① ( F ) := ✶ − β ♠✐♥ ( F ) · � i V ❛r [ f i ] ●❡♦♠ − β P♦✐ss ❇✐♥♦ β δ x , ✶ s s β −∞ ✵ ✶ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ❜♦♥✉s s✉r♣r✐s❡ ■❢ ②♦✉ ❤❛✈❡ t❤❛t g ( x ) ✐s ❇❡r♥ b ♦r ●❡♦♠ b ✱ t❤❡♥ t❤❡ ❇❇❍▲ ❛❧❣♦r✐t❤♠ ❛t ♣❛r❛♠❡t❡rs [ n , m ] ❝❛♥ ❜❡ ✉s❡❞ ✇❤❡♥ s❛♠♣❧✐♥❣ ❢r♦♠ � i g ∗ s i ( x i ) δ | x | , M ✱ ❜② r❡✇r✐t✐♥❣ x i = y ( ✶ ) + · · · + y ( s i ) i i ❈❛❧❧✐♥❣ s = � i s i ✱ ◮ ❇❇❍▲ [ s , M ] ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ g = ❇❡r♥ b ❝❛s❡✱ ❜② ✐❞❡♥t✐❢②✐♥❣ t❤❡ ♦✉t❝♦♠❡ str✐♥❣ ♦❢ ❇❇❍▲ ✇✐t❤ t❤❡ ❧✐st ♦❢ y ( j ) ✬s✳ i ◮ ❇❇❍▲ [ s + M − ✶ , M ] ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ g = ●❡♦♠ b ❝❛s❡✱ ❜② ✐❞❡♥t✐❢②✐♥❣ t❤❡ ❧❡♥❣t❤s ♦❢ r✉♥s ♦❢ ✵s ✐♥ t❤❡ ♦✉t❝♦♠❡ str✐♥❣ ♦❢ ❇❇❍▲ ✇✐t❤ t❤❡ ❧✐st ♦❢ y ( j ) ✬s✳ i ❚❤❡ P♦✐ss♦♥✐❛♥ ❝❛s❡ ✭♠♦r❡ s❡❧❞♦♠❧② ♥❡❡❞❡❞✮ ❝❛♥ ❜❡ ❞❡❛❧❡❞ ✇✐t❤ ❛ s♠❛❧❧ ❛❧❣♦r✐t❤♠ t❤❛t ■ ✐♥✈❡♥t❡❞✱ s✐♠✐❧❛r ✐♥ s♣✐r✐t t♦ ❇❇❍▲ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❊①❛♠♣❧❡s ♦❢ ❛♣♣❧✐❝❛t✐♦♥ ❙♦✱ ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ♦✉r ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❧✐♥❡❛r✲t✐♠❡ ❡①❛❝t s❛♠♣❧✐♥❣ ♦❢ s✉♠✲❝♦♥str❛✐♥❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐♥ t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡② ❛r❡ ♥♦t ❡q✉❛❧❧② ❞✐str✐❜✉t❡❞✳ ❍♦✇❡✈❡r✱ ②♦✉ ❝♦✉❧❞ ❥✉st t❤✐♥❦✿ ✓✇❤♦ ❝❛r❡s ❛❜♦✉t ♥♦t✲❡q✉❛❧❧②✲❞✐str✐❜✉t❡❞ ✈❛r✐❛❜❧❡s❄ ❆❢t❡r ❛❧❧✱ ❡✈❡r② t✐♠❡ ■ ✇❛♥t❡❞ t♦ ❣❡♥❡r❛t❡ ✇❛❧❦s✱ tr❡❡s✱ ❡t❝✳✱ ■ ❛❧✇❛②s ✇❛♥t❡❞ ❡q✉❛❧❧②✲❞✐str✐❜✉t❡❞ ✈❛r✐❛❜❧❡s✳ ✳ ✳ ✔ ❚❤❡ ♣♦✐♥t ✐s✿ ❡①❛♠♣❧❡s ♦❢ t❤✐s s♦rt ♠❛② ❜❡ ❤✐❞❞❡♥ ❜❡②♦♥❞ s♦♠❡ s♠❛rt ❜✐❥❡❝t✐♦♥✱ st❛rt✐♥❣ ❢r♦♠ ♠♦r❡ ❝✉st♦♠❛r② ✭❛♥❞ s②♠♠❡tr✐❝✮ ♣r♦❜❧❡♠s✳ ❚❤✐s ✐s ✇❡❧❧ ✐❧❧✉str❛t❡❞ ❜② t✇♦ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡s✿ • ❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ • P❡r♠✉t❛t✐♦♥s ✇✐t❤ m ❝②❝❧❡s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✜rst ❦✐♥❞ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❧t❤♦✉❣❤ t❤❡ s❡ts ❛r❡ ♥♦t ❧❛❜❡❧❡❞✱ t❤❡② ❛r❡ ❝❛♥♦♥✐❝❛❧❧② ♦r❞❡r❡❞✱ ❡✳❣✳ ❜② t❤❡✐r s♠❛❧❧❡st ❡❧❡♠❡♥t✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ❤❛✈❡ ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❝✐❞❡♥❝❡ ♠❛tr✐① ✱ ✇✐t❤ ✶ ✐❢ t❤❡ ❡❧❡♠❡♥t ✐s ✐♥ s✉❜s❡t ✳ ❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ❈❛❧❧ S s❡t n , m t❤❡ ❡♥s❡♠❜❧❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ ❛ s❡t ✇✐t❤ n ✭❧❛❜❡❧❧❡❞✮ ❡❧❡♠❡♥ts ✐♥t♦ m ✭✉♥❧❛❜❡❧❡❞✮ ♥♦♥✲❡♠♣t② s✉❜s❡ts✳ ❲✳❧✳♦✳❣✳ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ s❡t ❤❛s ❛ t♦t❛❧ ♦r❞❡r✐♥❣✳ ❊①❛♠♣❧❡✱ ❢♦r ( n , m ) = ( ✷✽ , ✾ ) ✱ ❛♥❞ t❤❡ s❡t { a , b , c , d , e , f , g , h , i , j , k , l , m , n , o , p , q , r , s , t , u , v , w , x , y , z , α, β } ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐t✐♦♥ � { a , g , t } , { b , d , m , o , α } , { c , j , s , y } , { e , h , v } , { f , k , q } , { i , l , z } , � { n , p , u } , { r , β } , { w , x } ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ❈❛❧❧ S s❡t n , m t❤❡ ❡♥s❡♠❜❧❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ ❛ s❡t ✇✐t❤ n ✭❧❛❜❡❧❧❡❞✮ ❡❧❡♠❡♥ts ✐♥t♦ m ✭✉♥❧❛❜❡❧❡❞✮ ♥♦♥✲❡♠♣t② s✉❜s❡ts✳ ❲✳❧✳♦✳❣✳ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ s❡t ❤❛s ❛ t♦t❛❧ ♦r❞❡r✐♥❣✳ ❊①❛♠♣❧❡✱ ❢♦r ( n , m ) = ( ✷✽ , ✾ ) ✱ ❛♥❞ t❤❡ s❡t { a , b , c , d , e , f , g , h , i , j , k , l , m , n , o , p , q , r , s , t , u , v , w , x , y , z , α, β } ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐t✐♦♥ � { a , g , t } , { b , d , m , o , α } , { c , j , s , y } , { e , h , v } , { f , k , q } , { i , l , z } , � { n , p , u } , { r , β } , { w , x } ❆❧t❤♦✉❣❤ t❤❡ s❡ts ❛r❡ ♥♦t ❧❛❜❡❧❡❞✱ t❤❡② ❛r❡ ❝❛♥♦♥✐❝❛❧❧② ♦r❞❡r❡❞✱ ❡✳❣✳ ❜② t❤❡✐r s♠❛❧❧❡st ❡❧❡♠❡♥t✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ❤❛✈❡ ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❝✐❞❡♥❝❡ ♠❛tr✐① T ✱ ✇✐t❤ T ij = ✶ ✐❢ t❤❡ ❡❧❡♠❡♥t j ✐s ✐♥ s✉❜s❡t i ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♥✉♠❜❡r ♦❢ ♣❛rt✐t✐♦♥s ✇✐t❤ ✐s t❤❡ tr✐✈✐❛❧ ♣r♦❞✉❝t✿ ✱ ❜✉t t❤❡ q✉❛♥t✐t✐❡s ✶ ❛r❡ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞✿ ✳ ❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ � { a , g , t } , { b , d , m , o , α } , { c , j , s , y } , { e , h , v } , { f , k , q } , { i , l , z } , � { n , p , u } , { r , β } , { w , x } ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ ✶ a b c d e f g h i j k l mn o p q r s t u v w x y z α β ❈❛❧❧ ❜❛❝❦❜♦♥❡ B ( T ) t❤❡ ❧✐st ♦❢ s♠❛❧❧❡st ❡❧❡♠❡♥ts ✐♥ t❤❡ s✉❜s❡ts✱ ❤❡r❡ B = { a , b , c , e , f , i , n , r , w } ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ � { a , g , t } , { b , d , m , o , α } , { c , j , s , y } , { e , h , v } , { f , k , q } , { i , l , z } , � { n , p , u } , { r , β } , { w , x } ✾ ✽ c y ✼ � �� � ✻ ✺ ✹ ✸ ✷ ✶ a b c d e f g h i j k l mn o p q r s t u v w x y z α β ❈❛❧❧ ❜❛❝❦❜♦♥❡ B ( T ) t❤❡ ❧✐st ♦❢ s♠❛❧❧❡st ❡❧❡♠❡♥ts ✐♥ t❤❡ s✉❜s❡ts✱ ❤❡r❡ B = { a , b , c , e , f , i , n , r , w } ✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❛rt✐t✐♦♥s T ✇✐t❤ B ( T ) = B ✐s t❤❡ tr✐✈✐❛❧ ♣r♦❞✉❝t✿ � m y = ✶ y c y ✱ ❜✉t t❤❡ q✉❛♥t✐t✐❡s c y ❛r❡ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞✿ � y c y = n − m ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ❆s ❛ r❡s✉❧t✱ s❛♠♣❧✐♥❣ ✉♥✐❢♦r♠❧② s❡t ♣❛rt✐t✐♦♥s ✐♥ S n , m ✱ ✇❤✐❝❤ ❜✐❥❡❝t✐✈❡❧② ❝♦✐♥❝✐❞❡s t♦ s❛♠♣❧✐♥❣ ✉♥✐❢♦r♠❧② t❤❡ t❛❜❧❡❛✉① T ✱ ❜♦✐❧s ❞♦✇♥ t♦ s❛♠♣❧✐♥❣ t❤❡ ❜❛❝❦❜♦♥❡ B ✇✐t❤ t❤❡ ♥♦♥✲✉♥✐❢♦r♠ ♠❡❛s✉r❡ µ n , m ( c ✶ , . . . , c m ) ∝ � m y = ✶ y c y × δ | c | , n − m ❚❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ■♥tr♦❞✉❝❡ ❛♥ ❛♣♣r♦♣r✐❛t❡ � y c y ✱ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ E ( | c | ) = n − m ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ω m = − ln( ✶ − ω ) ✭t❤❡ ❣♦♦❞ ❝❤♦✐❝❡ ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❡q✉❛t✐♦♥ n ✮ ω ω y ❚❤❡ ❢✉♥❝t✐♦♥s f y ( c y ) ❛r❡ ●❡♦♠ b y ( c y ) ✱ ✇✐t❤ b y = n − ω y ◆♦✇✱ ●❡♦♠ a ❤❛s ❛ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❡r♠s ♦❢ ❇❡r♥ b ●❡♦♠ a ( x ) = � a + b ( s ) ❇❡r♥ ∗ s s ●❡♦♠ b ( x ) a ❈❤♦♦s✐♥❣ ❢♦r s✐♠♣❧✐❝✐t② b = ✶ ✷ ✱ ♦✉r ❛❧❣♦r✐t❤♠ ✇♦r❦s✱ ✇✐t❤ ❛♥ � e − θ − ✶ + θ ✭ ω = ✶ − e − θ ✮ ❛✈❡r❛❣❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ E ( a ) = ✷ ( e θ − ✶ − θ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶ )( ✷ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶ )( ✷✸ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶ )( ✷✸ )( ✹ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶✺ )( ✷✸ )( ✹ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶✺ )( ✷✻✸ )( ✹ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶✺ )( ✷✻✸ )( ✹ )( ✼ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♥✉♠❜❡r ♦❢ ✬s ✇✐t❤ ❜❛❝❦❜♦♥❡ ✐s ✶ ✱ ✶ ❛♥❞ ✇❡ ♠✉st ❤❛✈❡ ① ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) ❈❛❧❧ B ( σ ) = { ✵ , ✵ , ✶ , ✵ , ✶ , ✶ , ✵ , ✶ } ✱ t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ ✏❜❧❛❝❦ r♦✇s✑ ♦❢ T ( σ ) ✱ t❤❡ ❜❛❝❦❜♦♥❡ ♦❢ σ ✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m ✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞ ❈❛❧❧ S ❝②❝ n , m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ S n ✇✐t❤ m ❝②❝❧❡s✳ ❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ � � ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) � � σ = ( ✶✺ )( ✷✻✸✽ )( ✹ )( ✼ ) ❈❛❧❧ B ( σ ) = { ✵ , ✵ , ✶ , ✵ , ✶ , ✶ , ✵ , ✶ } ✱ t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ ✏❜❧❛❝❦ r♦✇s✑ ♦❢ T ( σ ) ✱ t❤❡ ❜❛❝❦❜♦♥❡ ♦❢ σ ✳ ❚❤❡ ♥✉♠❜❡r ♦❢ σ ✬s ✇✐t❤ ❜❛❝❦❜♦♥❡ B = ( x ✶ , . . . , x n ) ✐s � y ( y − ✶ ) x y ✱ ❛♥❞ ✇❡ ♠✉st ❤❛✈❡ | ① | = m ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P❧❛♥ ♦❢ t❤❡ t❛❧❦ ❙♦✱ ■✬♠ tr②✐♥❣ t♦ ❵s❡❧❧✬ ②♦✉ t✇♦ r❡❝❡♥t t♦♦❧s ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✏❇♦❧t③♠❛♥♥✕❧✐❦❡✑ ♣r♦❜❧❡♠s ■ ✐♥tr♦❞✉❝❡❞✳ ■♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑✱ ✐❢ ✇❡ ❤❛✈❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ f i ✬s ✐♥ t❡r♠s ♦❢ ❛ ❢✉♥❝t✐♦♥ g ❜❡✐♥❣ ❇❡r♥ ♦r ●❡♦♠ ✱ ■ s❤♦✉❧❞ ❤❛✈❡ ❝♦♥✈✐♥❝❡❞ ②♦✉ t❤❛t ♠② tr✐❝❦ ✐s ❵♦♣t✐♠❛❧ ✉♣ t♦ ❛ ❢❛❝t♦r✬✳ ❨♦✉ s❤❛❧❧ ❜❡ ❤❛♣♣② ✇✐t❤ t❤✐s✱ ✉♥❧❡ss ②♦✉ r❡❛❧❧② s❡❛r❝❤ ❢♦r r❛♥❞✲❜✐t ♦♣t✐♠❛❧✐t② ✭❧✐❦❡ ✐♥ t❤❡ ♥✐❝❡ ❇❇❍▲ ❛❧❣♦r✐t❤♠✮✳ ❍♦✇❡✈❡r✱ ✇❡ ❛r❡ ♥♦t ❛❧✇❛②s s♦ ❧✉❝❦②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ✇❡ ❛r❡ ✐♥ t❤❡ ✏❇♦❧t③♠❛♥♥ ❝❛s❡✑✱ ✇❡ ❛❧r❡❛❞② st❛rt ❢r♦♠ ❛ ✇♦rs❡ ❝♦♠♣❧❡①✐t② ✭ ∼ N ✷ ❜❡❝❛✉s❡ ♦❢ ❢❛t t❛✐❧s✮✱ ❛♥❞ ✇❡ ❤❛✈❡ ♠✉❝❤ ❧❡ss t♦♦❧s ✐♥ ♦✉r ❤❛♥❞s✳ ✳ ✳ ■t✬s t✐♠❡ ❢♦r ♠❡ ❢♦r tr②✐♥❣ t♦ s❡❧❧ ②♦✉ t❤❡ s❡❝♦♥❞ t♦♦❧✿ t❤❡ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

▲❡t ❙❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✿ t❤❡ ♣r♦❜❧❡♠ ❲❡ ❤❛✈❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Z ✱ p ( x ) ❛♥❞ q ( x ) ✳ p ( x ) q ( x ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✿ t❤❡ ♣r♦❜❧❡♠ p ( x ) q ( x ) ❲❡ ❤❛✈❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Z ✱ p ( x ) ❛♥❞ q ( x ) ✳ ▲❡t f ( x ) = � y p ( y ) q ( y ) p ( x ) f ( x ) q ( x ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✿ t❤❡ ♣r♦❜❧❡♠ p ( x ) q ( x ) ❲❡ ❤❛✈❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Z ✱ p ( x ) ❛♥❞ q ( x ) ✳ ▲❡t f ( x ) = � y p ( y ) q ( y ) p ( x ) f ( x ) q ( x ) ❲❡ ❤❛✈❡ t✇♦ ❜❧❛❝❦✲❜♦① ❛❧❣♦r✐t❤♠s t❤❛t s❛♠♣❧❡ ❢r♦♠ p ❛♥❞ ❢r♦♠ q ✱ ❛♥❞ ✇❡ ✇❛♥t t♦ s❛♠♣❧❡ ❢r♦♠ f ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ❚❤❡ r✉❧❡s ♦❢ t❤❡ ❣❛♠❡ ❛r❡ ❝❧❡❛r✳ ❲❡ ❤❛✈❡ ♥♦ ❡①♣❧♦✐t❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ✇❤❛ts♦❡✈❡r ♦♥ p ❛♥❞ q ✳ ❲❡ ♦♥❧② ❤❛✈❡ t❤❡ ❜❧❛❝❦ ❜♦①❡s ❚❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ s❡❡♠s t♦ ❜❡ t❤❡ ♦♥❧② ❝❛♥❞✐❞❛t❡✿ ❆❧❣♦r✐t❤♠ ✿ ❖❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❜❡❣✐♥ r❡♣❡❛t x ⇐ p ; ② ⇐ q ✉♥t✐❧ x = y ❀ r❡t✉r♥ x ❡♥❞ ❉❡✜♥❡ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ( f , g ) = � x f ( x ) g ( x ) ✳ ❚❤❡ ❵r❡♣❡❛t✬ ❧♦♦♣ ✐s r❡♣❡❛t❡❞ ♦♥ ❛✈❡r❛❣❡ ✶ / ( p , q ) t✐♠❡s✳ ■❢ ✇❡ ❤❛✈❡ ❛ s✐③❡ ♣❛r❛♠❡t❡r n ✱ t❤✐s ♠❛② ❜❡ ❧❛r❣❡✱ ❛♥❞ ✇❡ ✇❛♥t t♦ ♠❛❦❡ ❜❡tt❡r ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❝❛❧✐♥❣ ♦❢ t❤❡ s✐③❡ ♣❛r❛♠❡t❡r x ( p ( x ) κ + q ( x ) κ ) ≪ ( p , q ) ❙❛② t❤❛t t❤❡r❡ ❡①✐sts ❛ κ ≥ ✸ s✉❝❤ t❤❛t � ✭❜② ❈❛✉❝❤②✕❙❝❤✇❛r③✱ ✐t ❝❛♥✬t ❜❡ κ = ✷✮✳ ❚❤❡♥ ✇❡ ✇❛♥t t♦ ❜r✐♥❣ ❞♦✇♥ t❤❡ ❝♦♠♣❧❡①✐t② � ❢r♦♠ ∼ ✶ / ( p , q ) ✳ t♦ ∼ ✶ / ( p , q ) ✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡ t②♣✐❝❛❧ ❝❛s❡ ✐s κ = ✸ ❛❜♦✈❡✱ ❛♥❞ ( p , p ) ✱ ( p , q ) ✱ ( q , q ) ❛r❡ ❛❧❧ Θ( n − ✷ α ) ✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s Θ( n ✷ α ) ❛♥❞ ✇❡ ✇❛♥t t♦ ❣♦ ❞♦✇♥ t♦ Θ( n α ln n ) ♦r Θ( n α ) Θ( n ✷ α ) ✛ ✲ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❝❛❧✐♥❣ ♦❢ t❤❡ s✐③❡ ♣❛r❛♠❡t❡r x ( p ( x ) κ + q ( x ) κ ) ≪ ( p , q ) ❙❛② t❤❛t t❤❡r❡ ❡①✐sts ❛ κ ≥ ✸ s✉❝❤ t❤❛t � ✭❜② ❈❛✉❝❤②✕❙❝❤✇❛r③✱ ✐t ❝❛♥✬t ❜❡ κ = ✷✮✳ ❚❤❡♥ ✇❡ ✇❛♥t t♦ ❜r✐♥❣ ❞♦✇♥ t❤❡ ❝♦♠♣❧❡①✐t② � ❢r♦♠ ∼ ✶ / ( p , q ) ✳ t♦ ∼ ✶ / ( p , q ) ✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡ t②♣✐❝❛❧ ❝❛s❡ ✐s κ = ✸ ❛❜♦✈❡✱ ❛♥❞ ( p , p ) ✱ ( p , q ) ✱ ( q , q ) ❛r❡ ❛❧❧ Θ( n − ✷ α ) ✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s Θ( n ✷ α ) ❛♥❞ ✇❡ ✇❛♥t t♦ ❣♦ ❞♦✇♥ t♦ Θ( n α ln n ) ♦r Θ( n α ) Θ( n ✷ α ) ✛ ✲ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ❝♦♠♣❧❡①✐t② ♣❛r❛❞✐❣♠ ✭②♦✉ ♥❡❡❞ t❤✐s ♦♥❧② ✐❢ ②♦✉ ❝❛r❡ ❛❜♦✉t ❧♦❣s✮ ■♥ ❛❧❣♦r✐t❤♠ ❝♦♠♣❧❡①✐t② ②♦✉ ♥♦r♠❛❧❧② ❥✉st ❝♦✉♥t ♦♣❡r❛t✐♦♥s✳ ◆♦♥❡t❤❡❧❡ss✱ ✇❤❡♥ ②♦✉ ❤❛✈❡ ❜❧❛❝❦ ❜♦①❡s✱ ✐t ✐s ✇✐s❡ t♦ ❝♦✉♥t s❡♣❛r❛t❡❧② ♦♣❡r❛t✐♦♥s ❛♥❞ ❜❧❛❝❦✲❜♦① q✉❡r✐❡s✱ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛❧❧② ♠✉❝❤ ♠♦r❡ ❡①♣❡♥s✐✈❡ t❤❛♥ ❛ s✐♥❣❧❡ ♦♣❡r❛t✐♦♥ ❍❡r❡ ✇❡ ❤❛✈❡ t✇♦ ❜❧❛❝❦ ❜♦①❡s✱ ❢♦r s✐♠♣❧✐❝✐t② ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❡② ❤❛✈❡ s✐♠✐❧❛r ❝♦♠♣❧❡①✐t② � ❢♦r ❜❧❛❝❦✲❜♦① q✉❡r✐❡s ◆✐❝❡ ♥♦t❛t✐♦♥✿ ❢♦r ♦♣❡r❛t✐♦♥s ❚❤❡♥✱ ❡✳❣✳✱ Θ( n x + n y ) s✐♠♣❧✐✜❡s t♦ Θ( n x ) ✐❢ x ≥ y ✱ ❜✉t st❛②s ❛s ✐s ✐❢ x < y ✱ ❛s ✇❡ ♦♥❧② ❦♥♦✇ t❤❛t / > ✶ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s Θ( n ✷ α ) + n α ln( n ) ❲❡ ✇❛♥t t♦ ❣♦ ❞♦✇♥ t♦ Θ( n α ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♥❛ï✈❡ ❵❜✐rt❤❞❛② ♣❛r❛❞♦①✬ ❛❧❣♦r✐t❤♠ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ❆❧❣♦r✐t❤♠ ✿ ❇✐rt❤❞❛② ♣❛r❛❞♦①✱ ✜rst tr② ❜❡❣✐♥ r❡♣❡❛t ( x ✶ , . . . , x k ) ⇐ p ; ✭② ✶ , . . . , y k ) ⇐ q ✉♥t✐❧ ∃ ! ( i , j ) | x i = y j ❀ r❡t✉r♥ x i ❡♥❞ ❇❡st ❤♦♣❡✿ ✐♥ ❡❛❝❤ ❵r❡♣❡❛t✬ ❝②❝❧❡ t❤❡r❡ ❛r❡ ∼ P♦✐ss k ✷ ( p , q ) ♣❛✐rs ( i , j ) s✉❝❤ t❤❛t x i = y j ✱ � s♦ ✐❢ ✇❡ t✉♥❡ k ∼ ✶ / ( p , q ) t❤❡ ❝②❝❧❡ ❝♦sts Θ( k ) ✱ ❛♥❞ ✐s r❡♣❡❛t❡❞ ♦♥ ❛✈❡r❛❣❡ Θ( ✶ ) t✐♠❡s ❚❤❡ ❝♦♠♣❧❡①✐t② ❞r♦♣s ❞♦✇♥ t♦ Θ( n α ) ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

✳ ✳ ✳ ◆♦✦ ❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s ✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✷ ✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡ t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✇✐t❤ ✷ ✱ ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ ✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r ✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳ ❆♥ ❡❛s② ✇✐♥❄ ❆♥ ❡❛s② ✇✐♥❄ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s ✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✷ ✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡ t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✇✐t❤ ✷ ✱ ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ ✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r ✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳ ❆♥ ❡❛s② ✇✐♥❄ ❆♥ ❡❛s② ✇✐♥❄ ✳ ✳ ✳ ◆♦✦ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣