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s t r rt St ( p ) str tr P trs ss t t


  1. ✐s t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ✳ St ( p ) ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❆❧❢ ❖♥s❤✉✉s ❉❡✜♥✐t✐♦♥ ❙t✭♣✮ ❋✐① µ ❜❡ ❛♥ A ✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ ❞❡✜♥❛❜❧❡ s❡ts✳ ❚❤❡♥ St ( p ) ✐s ❚❤❡ ❙t❛❜✐❧✐③❡r t❤❡ s❡t ♦❢ a s✉❝❤ t❤❛t ap ( x ) ∩ p ( x ) ✐s ♥♦t ✐♥ µ ✳ ❚❤❡♦r❡♠ ❊q✉✐✈❛❧❡♥t❧②✱ a ✐s ✐♥ St ( p ) ✐❢ t❤❡r❡ ✐s b | = p s✉❝❤ t❤❛t tp ( b / Aa ) ✐s µ ✲❧❛r❣❡ ❛♥❞ ab | = p ✳

  2. St ( p ) ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❆❧❢ ❖♥s❤✉✉s ❉❡✜♥✐t✐♦♥ ❙t✭♣✮ ❋✐① µ ❜❡ ❛♥ A ✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ ❞❡✜♥❛❜❧❡ s❡ts✳ ❚❤❡♥ St ( p ) ✐s ❚❤❡ ❙t❛❜✐❧✐③❡r t❤❡ s❡t ♦❢ a s✉❝❤ t❤❛t ap ( x ) ∩ p ( x ) ✐s ♥♦t ✐♥ µ ✳ ❚❤❡♦r❡♠ ❊q✉✐✈❛❧❡♥t❧②✱ a ✐s ✐♥ St ( p ) ✐❢ t❤❡r❡ ✐s b | = p s✉❝❤ t❤❛t tp ( b / Aa ) ✐s µ ✲❧❛r❣❡ ❛♥❞ ab | = p ✳ Stab ( p ) ✐s t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② St ( p ) ✳

  3. ❖❜s❡r✈❛t✐♦♥ ❆♥② ❢♦r♠✉❧❛ t❤❛t ❞✐✈✐❞❡s ✭♦r ❢♦r❦s✮ ♦✈❡r ♠✉st ❜❡❧♦♥❣ t♦ ❛♥② ✶ ✲❆✲✐♥✈❛r✐❛♥t ✐❞❡❛❧✳ ❚❤❡ ❙✶ Pr♦♣❡rt② ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❉❡✜♥✐t✐♦♥ ❆❧❢ ❖♥s❤✉✉s ▲❡t µ ❜❡ ❛♥ A ✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ ❞❡✜♥❛❜❧❡ s❡ts✳ ❘❡❝❛❧❧ t❤❛t µ ❙t✭♣✮ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt② ✐❢ ❣✐✈❡♥ ❛♥② φ ( x , a ✵ ) ❛♥❞ φ ( x , a ✶ ) ✱ ❢♦r a ✵ ❚❤❡ ❛♥❞ a ✶ st❛rt✐♥❣ ❛♥ ✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡✱ ✐❢ φ ( x , a ✵ ) ∧ φ ( x , a ✶ ) ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ✐s ✐♥ µ t❤❡♥ s♦ ✐s φ ( x , a ✵ ) ✳

  4. ❚❤❡ ❙✶ Pr♦♣❡rt② ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❉❡✜♥✐t✐♦♥ ❆❧❢ ❖♥s❤✉✉s ▲❡t µ ❜❡ ❛♥ A ✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ ❞❡✜♥❛❜❧❡ s❡ts✳ ❘❡❝❛❧❧ t❤❛t µ ❙t✭♣✮ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt② ✐❢ ❣✐✈❡♥ ❛♥② φ ( x , a ✵ ) ❛♥❞ φ ( x , a ✶ ) ✱ ❢♦r a ✵ ❚❤❡ ❛♥❞ a ✶ st❛rt✐♥❣ ❛♥ ✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡✱ ✐❢ φ ( x , a ✵ ) ∧ φ ( x , a ✶ ) ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ✐s ✐♥ µ t❤❡♥ s♦ ✐s φ ( x , a ✵ ) ✳ ❖❜s❡r✈❛t✐♦♥ ❆♥② ❢♦r♠✉❧❛ t❤❛t ❞✐✈✐❞❡s ✭♦r ❢♦r❦s✮ ♦✈❡r A ♠✉st ❜❡❧♦♥❣ t♦ ❛♥② S ✶ ✲❆✲✐♥✈❛r✐❛♥t ✐❞❡❛❧✳

  5. Pr♦♦❢✳ ❙✉♣♣♦s❡ ♥♦t✳ ▲❡t ✲✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡ ♣r♦✈✐♥❣ ✐t ✐s ♥♦t st❛❜❧❡✳ ✶ ✐s ✇✐❞❡ ✐✛ ✶ ❙♦ ✳ ✶ ✐s ✇✐❞❡✱ ❜✉t ✶ ❇✉t t❤❡♥ ✶ ✐s ♥♦t✱ ❛ ✶ ✶ ✶ ✶ ✶ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❈❧❛✐♠ ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ▲❡t p ❜❡ ❛ t②♣❡ ❛♥❞ ❛ss✉♠❡ t❤❛t µ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt②✳ ❚❤❡♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❢♦r ❛♥② t②♣❡ q t❤❡ r❡❧❛t✐♦♥ ❆❧❢ ❖♥s❤✉✉s ⇒ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s µ ✲✇✐❞❡ , R ( a , b ) ⇐ ❙t✭♣✮ ❚❤❡ ✇❤❡r❡ ✇❡ ✐❞❡♥t✐❢② ❛ t②♣❡ ✇✐t❤ ✐ts r❡❛❧✐③❛t✐♦♥s ✐♥ t❤❡ ♠♦♥st❡r ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ♠♦❞❡❧✱ ✐s ❛ st❛❜❧❡ r❡❧❛t✐♦♥✳

  6. ✶ ✐s ✇✐❞❡ ✐✛ ✶ ❙♦ ✳ ✶ ✐s ✇✐❞❡✱ ❜✉t ✶ ❇✉t t❤❡♥ ✶ ✐s ♥♦t✱ ❛ ✶ ✶ ✶ ✶ ✶ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❈❧❛✐♠ ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ▲❡t p ❜❡ ❛ t②♣❡ ❛♥❞ ❛ss✉♠❡ t❤❛t µ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt②✳ ❚❤❡♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❢♦r ❛♥② t②♣❡ q t❤❡ r❡❧❛t✐♦♥ ❆❧❢ ❖♥s❤✉✉s ⇒ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s µ ✲✇✐❞❡ , R ( a , b ) ⇐ ❙t✭♣✮ ❚❤❡ ✇❤❡r❡ ✇❡ ✐❞❡♥t✐❢② ❛ t②♣❡ ✇✐t❤ ✐ts r❡❛❧✐③❛t✐♦♥s ✐♥ t❤❡ ♠♦♥st❡r ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ♠♦❞❡❧✱ ✐s ❛ st❛❜❧❡ r❡❧❛t✐♦♥✳ Pr♦♦❢✳ • ❙✉♣♣♦s❡ ♥♦t✳ ▲❡t � ( a i , b i ) � i ∈ ω A ✲✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡ ♣r♦✈✐♥❣ ✐t ✐s ♥♦t st❛❜❧❡✳

  7. ✶ ✐s ✇✐❞❡✱ ❜✉t ✶ ❇✉t t❤❡♥ ✶ ✐s ♥♦t✱ ❛ ✶ ✶ ✶ ✶ ✶ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❈❧❛✐♠ ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ▲❡t p ❜❡ ❛ t②♣❡ ❛♥❞ ❛ss✉♠❡ t❤❛t µ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt②✳ ❚❤❡♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❢♦r ❛♥② t②♣❡ q t❤❡ r❡❧❛t✐♦♥ ❆❧❢ ❖♥s❤✉✉s ⇒ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s µ ✲✇✐❞❡ , R ( a , b ) ⇐ ❙t✭♣✮ ❚❤❡ ✇❤❡r❡ ✇❡ ✐❞❡♥t✐❢② ❛ t②♣❡ ✇✐t❤ ✐ts r❡❛❧✐③❛t✐♦♥s ✐♥ t❤❡ ♠♦♥st❡r ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ♠♦❞❡❧✱ ✐s ❛ st❛❜❧❡ r❡❧❛t✐♦♥✳ Pr♦♦❢✳ • ❙✉♣♣♦s❡ ♥♦t✳ ▲❡t � ( a i , b i ) � i ∈ ω A ✲✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡ ♣r♦✈✐♥❣ ✐t ✐s ♥♦t st❛❜❧❡✳ • ❙♦ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s ✇✐❞❡ ✐✛ i ≤ j ✳ i j

  8. ❈❧❛✐♠ ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ▲❡t p ❜❡ ❛ t②♣❡ ❛♥❞ ❛ss✉♠❡ t❤❛t µ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt②✳ ❚❤❡♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❢♦r ❛♥② t②♣❡ q t❤❡ r❡❧❛t✐♦♥ ❆❧❢ ❖♥s❤✉✉s ⇒ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s µ ✲✇✐❞❡ , R ( a , b ) ⇐ ❙t✭♣✮ ❚❤❡ ✇❤❡r❡ ✇❡ ✐❞❡♥t✐❢② ❛ t②♣❡ ✇✐t❤ ✐ts r❡❛❧✐③❛t✐♦♥s ✐♥ t❤❡ ♠♦♥st❡r ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ♠♦❞❡❧✱ ✐s ❛ st❛❜❧❡ r❡❧❛t✐♦♥✳ Pr♦♦❢✳ • ❙✉♣♣♦s❡ ♥♦t✳ ▲❡t � ( a i , b i ) � i ∈ ω A ✲✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡ ♣r♦✈✐♥❣ ✐t ✐s ♥♦t st❛❜❧❡✳ • ❙♦ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s ✇✐❞❡ ✐✛ i ≤ j ✳ i j • ❇✉t t❤❡♥ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s ✇✐❞❡✱ ❜✉t i i p ( x ) a − ✶ i + ✶ ∩ q ( x ) b − ✶ i + ✶ ∩ p ( x ) a − ✶ ∩ q ( x ) b − ✶ ✐s ♥♦t✱ ❛ i i ❝♦♥tr❛❞✐❝t✐♦♥✳

  9. ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❆❧❢ ❖♥s❤✉✉s ❙t✭♣✮ ❚❤❡♦r❡♠ ❚❤❡ ❙t❛❜✐❧✐③❡r ▲❡t µ ❜❡ ♠❡❞✐✉♠ ✐♥ p , q ✳ ❚❤❡♥ t❤❡ r❡❧❛t✐♦♥ R ( g , h ) ❞❡✜♥❡❞ ❛s ❚❤❡♦r❡♠ ✏ gp ∩ hq ✐s ✇✐❞❡✧ ✐s ❛ st❛❜❧❡ r❡❧❛t✐♦♥✳

  10. ❚❤❡ st❛❜✐❧✐③❡r t❤❡♦r❡♠ ✐♥ ◆❚P✷ t❤❡♦r✐❡s✳ ❆❧❢ ❖♥s❤✉✉s ❋❛❝t ✭❍r✉s❤♦✈s❦✐✮ ❙t✭♣✮ ▲❡t p , q ❜❡ ❝♦♠♣❧❡t❡ t②♣❡s ♦✈❡r ❛ ♠♦❞❡❧ M ❛♥❞ ❧❡t R ( x , y ) ❜❡ ❛ ❚❤❡ ❙t❛❜✐❧✐③❡r st❛❜❧❡ M ✲✐♥✈❛r✐❛♥t r❡❧❛t✐♦♥ ✐♥ t❤❡ r❡❛❧✐③❛t✐♦♥s ♦❢ p ( x ) × q ( y ) ✳ ❚❤❡♦r❡♠ ❚❤❡♥ t❤❡ tr✉t❤ ✈❛❧✉❡ ♦❢ R ( a , b ) ✐s ❝♦♥st❛♥t ❢♦r ❛❧❧ a | = p ( x ) ❛♥❞ b | = q ( y ) ❛s ❧♦♥❣ ❛s ❡✐t❤❡r tp ( a / Mb ) ♦r tp ( b / Ma ) ❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r M ✳

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