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❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u < d

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r

❑✉rt ●ö❞❡❧ ❘❡s❡❛r❝❤ ❈❡♥t❡r ❢♦r ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝✱ ❯♥✐✈❡rs✐t② ♦❢ ❱✐❡♥♥❛

▲♦❣✐❝ ❈♦❧❧♦q✉✐✉♠ ✷✵✵✼ ❲r♦❝➟❛✇ ❏✉❧② ✶✸ ✕ ✷✵✱ ✷✵✵✼

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❖✉t❧✐♥❡

✶ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ✷ ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ✸ ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ✹ ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

▼❛♣♣✐♥❣s ❜❡t✇❡❡♥ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❆ ✜❧t❡r ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♣r♦♣❡r ✜❧t❡r ♦♥ ω✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❢ : ω → ω ❜❡ ✜♥✐t❡✲t♦✲♦♥❡✳ ❲❡ s❡t ❢ (F) = {❳ : ❢ −✶❳ ∈ F}✳ ❢ ❝♦♥t❛✐♥s ❧❡ss ✐♥❢♦r♠❛t✐♦♥ t❤❛♥ ✿ ❢

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

▼❛♣♣✐♥❣s ❜❡t✇❡❡♥ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❆ ✜❧t❡r ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♣r♦♣❡r ✜❧t❡r ♦♥ ω✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❢ : ω → ω ❜❡ ✜♥✐t❡✲t♦✲♦♥❡✳ ❲❡ s❡t ❢ (F) = {❳ : ❢ −✶❳ ∈ F}✳ ❢ (F) ❝♦♥t❛✐♥s ❧❡ss ✐♥❢♦r♠❛t✐♦♥ t❤❛♥ F✿

✲ F

• ✠

✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✾ ✲ ❢ (F)

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

▼❛♣♣✐♥❣s ❜❡t✇❡❡♥ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❆ ✜❧t❡r ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♣r♦♣❡r ✜❧t❡r ♦♥ ω✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❢ : ω → ω ❜❡ ✜♥✐t❡✲t♦✲♦♥❡✳ ❲❡ s❡t ❢ (F) = {❳ : ❢ −✶❳ ∈ F}✳ ❢ (F) ❝♦♥t❛✐♥s ❧❡ss ✐♥❢♦r♠❛t✐♦♥ t❤❛♥ F✿

✲ F

• ✠

✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✾ ✲ ❢ (F)

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (F) ∪ ❢ (G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ (U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❚✇♦ ✉❧tr❛✜❧t❡rs ❛♥❞ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ s✉❝❤ t❤❛t ❢ ❢ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (F) ∪ ❢ (G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ (U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❚✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (U ) = ❢ (V )✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (F) ∪ ❢ (G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ (U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❚✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (U ) = ❢ (V )✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (F) ∪ ❢ (G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ (U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❚✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ (U ) = ❢ (V )✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✉❧tr❛✜❧t❡rs

■❢ ❢ (U ) = ❢ (V ) ❛♥❞ ❣(V ) = ❣(W )✱ t❤❡♥ t❤❡r❡ ✐s ❛ s❧♦✇❡r ❣r♦✇✐♥❣ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❤ s✉❝❤ t❤❛t ❤(U ) = ❤(W )✳ ❋❛❝t ❚❤❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ r❡❧❛t✐♦♥ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ t❤❡ ✉❧tr❛✜❧t❡rs ♦♥ ω✳ ■ts ❝❧❛ss❡s ❛r❡ ❝❛❧❧❡❞ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚✇♦ ✜❧t❡rs ❛♥❞ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐✛ t❤❡r❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❛♥❞ s✉❝❤ t❤❛t ❛♥❞ ✳ ❙♦ ✏◆❈❯✑ ✐♠♣❧✐❡s ◆❈❋✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✉❧tr❛✜❧t❡rs

■❢ ❢ (U ) = ❢ (V ) ❛♥❞ ❣(V ) = ❣(W )✱ t❤❡♥ t❤❡r❡ ✐s ❛ s❧♦✇❡r ❣r♦✇✐♥❣ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❤ s✉❝❤ t❤❛t ❤(U ) = ❤(W )✳ ❋❛❝t ❚❤❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ r❡❧❛t✐♦♥ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ t❤❡ ✉❧tr❛✜❧t❡rs ♦♥ ω✳ ■ts ❝❧❛ss❡s ❛r❡ ❝❛❧❧❡❞ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚✇♦ ✜❧t❡rs ❛♥❞ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐✛ t❤❡r❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❛♥❞ s✉❝❤ t❤❛t ❛♥❞ ✳ ❙♦ ✏◆❈❯✑ ✐♠♣❧✐❡s ◆❈❋✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✉❧tr❛✜❧t❡rs

■❢ ❢ (U ) = ❢ (V ) ❛♥❞ ❣(V ) = ❣(W )✱ t❤❡♥ t❤❡r❡ ✐s ❛ s❧♦✇❡r ❣r♦✇✐♥❣ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❤ s✉❝❤ t❤❛t ❤(U ) = ❤(W )✳ ❋❛❝t ❚❤❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ r❡❧❛t✐♦♥ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ t❤❡ ✉❧tr❛✜❧t❡rs ♦♥ ω✳ ■ts ❝❧❛ss❡s ❛r❡ ❝❛❧❧❡❞ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐✛ t❤❡r❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs U ❛♥❞ V s✉❝❤ t❤❛t U ⊇ F ❛♥❞ V ⊇ G ✳ ❙♦ ✏◆❈❯✑ ✐♠♣❧✐❡s ◆❈❋✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✉❧tr❛✜❧t❡rs

■❢ ❢ (U ) = ❢ (V ) ❛♥❞ ❣(V ) = ❣(W )✱ t❤❡♥ t❤❡r❡ ✐s ❛ s❧♦✇❡r ❣r♦✇✐♥❣ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❤ s✉❝❤ t❤❛t ❤(U ) = ❤(W )✳ ❋❛❝t ❚❤❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ r❡❧❛t✐♦♥ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ t❤❡ ✉❧tr❛✜❧t❡rs ♦♥ ω✳ ■ts ❝❧❛ss❡s ❛r❡ ❝❛❧❧❡❞ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐✛ t❤❡r❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs U ❛♥❞ V s✉❝❤ t❤❛t U ⊇ F ❛♥❞ V ⊇ G ✳ ❙♦ ✏◆❈❯✑ ✐♠♣❧✐❡s ◆❈❋✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

P♦ss✐❜❧❡ ♥✉♠❜❡rs ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

❚❤❡♦r❡♠✳ ❇♦♦t❤✱ ●❛❧✈✐♥✱ ▼❛r②✲❊❧❧❡♥ ❘✉❞✐♥✱ ❇❧❛ss ❯♥❞❡r ❈❍✱ t❤❡r❡ ❛r❡ ✷✷ω ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ❙❤❡❧❛❤✱ ✶✾✽✼ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡ ❢❛❝t✱ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✭✉❧tr❛✮✜❧t❡rs✱ ◆❈❋✳ ❈♦♥❥❡❝t✉r❡✿ ❚❤❡r❡ ✐s ❛ ♠♦❞❡❧ ✇✐t❤ ❡①❛❝t❧② t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❇✐❣ ♦♣❡♥ q✉❡st✐♦♥✿ ❖t❤❡r ✜♥✐t❡ ♥✉♠❜❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

P♦ss✐❜❧❡ ♥✉♠❜❡rs ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

❚❤❡♦r❡♠✳ ❇♦♦t❤✱ ●❛❧✈✐♥✱ ▼❛r②✲❊❧❧❡♥ ❘✉❞✐♥✱ ❇❧❛ss ❯♥❞❡r ❈❍✱ t❤❡r❡ ❛r❡ ✷✷ω ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ❙❤❡❧❛❤✱ ✶✾✽✼ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡ ❢❛❝t✱ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✭✉❧tr❛✮✜❧t❡rs✱ ◆❈❋✳ ❈♦♥❥❡❝t✉r❡✿ ❚❤❡r❡ ✐s ❛ ♠♦❞❡❧ ✇✐t❤ ❡①❛❝t❧② t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❇✐❣ ♦♣❡♥ q✉❡st✐♦♥✿ ❖t❤❡r ✜♥✐t❡ ♥✉♠❜❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

P♦ss✐❜❧❡ ♥✉♠❜❡rs ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

❚❤❡♦r❡♠✳ ❇♦♦t❤✱ ●❛❧✈✐♥✱ ▼❛r②✲❊❧❧❡♥ ❘✉❞✐♥✱ ❇❧❛ss ❯♥❞❡r ❈❍✱ t❤❡r❡ ❛r❡ ✷✷ω ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ❙❤❡❧❛❤✱ ✶✾✽✼ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡ ❢❛❝t✱ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✭✉❧tr❛✮✜❧t❡rs✱ ◆❈❋✳ ❈♦♥❥❡❝t✉r❡✿ ❚❤❡r❡ ✐s ❛ ♠♦❞❡❧ ✇✐t❤ ❡①❛❝t❧② t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❇✐❣ ♦♣❡♥ q✉❡st✐♦♥✿ ❖t❤❡r ✜♥✐t❡ ♥✉♠❜❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

P♦ss✐❜❧❡ ♥✉♠❜❡rs ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

❚❤❡♦r❡♠✳ ❇♦♦t❤✱ ●❛❧✈✐♥✱ ▼❛r②✲❊❧❧❡♥ ❘✉❞✐♥✱ ❇❧❛ss ❯♥❞❡r ❈❍✱ t❤❡r❡ ❛r❡ ✷✷ω ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ❙❤❡❧❛❤✱ ✶✾✽✼ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡ ❢❛❝t✱ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✭✉❧tr❛✮✜❧t❡rs✱ ◆❈❋✳ ❈♦♥❥❡❝t✉r❡✿ ❚❤❡r❡ ✐s ❛ ♠♦❞❡❧ ✇✐t❤ ❡①❛❝t❧② t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❇✐❣ ♦♣❡♥ q✉❡st✐♦♥✿ ❖t❤❡r ✜♥✐t❡ ♥✉♠❜❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

P♦ss✐❜❧❡ ♥✉♠❜❡rs ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

❚❤❡♦r❡♠✳ ❇♦♦t❤✱ ●❛❧✈✐♥✱ ▼❛r②✲❊❧❧❡♥ ❘✉❞✐♥✱ ❇❧❛ss ❯♥❞❡r ❈❍✱ t❤❡r❡ ❛r❡ ✷✷ω ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ❙❤❡❧❛❤✱ ✶✾✽✼ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ♦❢ ✉❧tr❛✜❧t❡rs✳ ❚❤❡ ❢❛❝t✱ t❤❛t t❤❡r❡ ✐s ❥✉st ♦♥❡ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✭✉❧tr❛✮✜❧t❡rs✱ ◆❈❋✳ ❈♦♥❥❡❝t✉r❡✿ ❚❤❡r❡ ✐s ❛ ♠♦❞❡❧ ✇✐t❤ ❡①❛❝t❧② t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❇✐❣ ♦♣❡♥ q✉❡st✐♦♥✿ ❖t❤❡r ✜♥✐t❡ ♥✉♠❜❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❊①❝❧✉❞❡❞ ♥✉♠❜❡rs

❚❤❡♦r❡♠✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ■❢ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs t❤❡♥ t❤❡r❡ ❛r❡ ✷✷ω ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ✶✾✽✼ d ≤ u ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ◗✉❡st✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ❉♦❡s ✐♠♣❧② t❤❛t t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs❄

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❊①❝❧✉❞❡❞ ♥✉♠❜❡rs

❚❤❡♦r❡♠✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ■❢ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs t❤❡♥ t❤❡r❡ ❛r❡ ✷✷ω ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ✶✾✽✼ d ≤ u ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ◗✉❡st✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ❉♦❡s u < d ✐♠♣❧② t❤❛t t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs❄

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❊①❝❧✉❞❡❞ ♥✉♠❜❡rs

❚❤❡♦r❡♠✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ■❢ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs t❤❡♥ t❤❡r❡ ❛r❡ ✷✷ω ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ✶✾✽✼ d ≤ u ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ◗✉❡st✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ❉♦❡s u < d ✐♠♣❧② t❤❛t t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs❄

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇❛s❡s ❛♥❞ ❝❤❛r❛❝t❡rs✱ u

❉❡✜♥✐t✐♦♥ ❆ s❡t B ⊆ F ✐s ❝❛❧❧❡❞ ❛ ❜❛s❡ ❢♦r F ✐❢ (∀❋ ∈ F)(∃❇ ∈ B)(❇ ⊆ ❋)✳ ❆ s❡t B ⊆ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ♣s❡✉❞♦❜❛s❡ ❢♦r F ✐❢ (∀❋ ∈ F)(∃❇ ∈ B)(❇ ⊆ ❋)✳ ❚❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ ❜❛s❡ ♦❢ ✐s ❝❛❧❧❡❞ ✱ t❤❡ ❝❤❛r❛❝t❡r ♦❢ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇❛s❡s ❛♥❞ ❝❤❛r❛❝t❡rs✱ u

❉❡✜♥✐t✐♦♥ ❆ s❡t B ⊆ F ✐s ❝❛❧❧❡❞ ❛ ❜❛s❡ ❢♦r F ✐❢ (∀❋ ∈ F)(∃❇ ∈ B)(❇ ⊆ ❋)✳ ❆ s❡t B ⊆ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ♣s❡✉❞♦❜❛s❡ ❢♦r F ✐❢ (∀❋ ∈ F)(∃❇ ∈ B)(❇ ⊆ ❋)✳ ❚❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ ❜❛s❡ ♦❢ F ✐s ❝❛❧❧❡❞ χ(F)✱ t❤❡ ❝❤❛r❛❝t❡r ♦❢ F✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇❛s❡s ❛♥❞ ❝❤❛r❛❝t❡rs✱ u

❉❡✜♥✐t✐♦♥ ❆ s❡t B ⊆ F ✐s ❝❛❧❧❡❞ ❛ ❜❛s❡ ❢♦r F ✐❢ (∀❋ ∈ F)(∃❇ ∈ B)(❇ ⊆ ❋)✳ ❆ s❡t B ⊆ [ω]ω ✐s ❝❛❧❧❡❞ ❛ ♣s❡✉❞♦❜❛s❡ ❢♦r F ✐❢ (∀❋ ∈ F)(∃❇ ∈ B)(❇ ⊆ ❋)✳ ❚❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ ❜❛s❡ ♦❢ F ✐s ❝❛❧❧❡❞ χ(F)✱ t❤❡ ❝❤❛r❛❝t❡r ♦❢ F✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❛❦✐♥❣ t❤❡ ♠✐♥✐♠✉♠ ♦✈❡r ❛❧❧ ✉❧tr❛✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚❤❡ ✉❧tr❛✜❧t❡r ❝❤❛r❛❝t❡r✐st✐❝ u ✐s t❤❡ ♠✐♥✐♠❛❧ χ(U ) ❢♦r ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r U ✳ ❚❤❡ r❡❛♣✐♥❣ ♥✉♠❜❡r r ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ♣s❡✉❞♦❜❛s❡ ❢♦r ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r U ✳ ❚❤❡♦r❡♠✱ ●♦❧❞st❡r♥✱ ❙❤❡❧❛❤✱ ✶✾✾✵ ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈✳ ❇✉t t❤❡♥ ❜② ❛ t❤❡♦r❡♠ ♦❢ ❆✉❜r❡②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❛❦✐♥❣ t❤❡ ♠✐♥✐♠✉♠ ♦✈❡r ❛❧❧ ✉❧tr❛✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚❤❡ ✉❧tr❛✜❧t❡r ❝❤❛r❛❝t❡r✐st✐❝ u ✐s t❤❡ ♠✐♥✐♠❛❧ χ(U ) ❢♦r ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r U ✳ ❚❤❡ r❡❛♣✐♥❣ ♥✉♠❜❡r r ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ♣s❡✉❞♦❜❛s❡ ❢♦r ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r U ✳ ❚❤❡♦r❡♠✱ ●♦❧❞st❡r♥✱ ❙❤❡❧❛❤✱ ✶✾✾✵ r < u ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈✳ ❇✉t t❤❡♥ d ≤ u ❜② ❛ t❤❡♦r❡♠ ♦❢ ❆✉❜r❡②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❛❦✐♥❣ t❤❡ ♠✐♥✐♠✉♠ ♦✈❡r ❛❧❧ ✉❧tr❛✜❧t❡rs

❉❡✜♥✐t✐♦♥ ❚❤❡ ✉❧tr❛✜❧t❡r ❝❤❛r❛❝t❡r✐st✐❝ u ✐s t❤❡ ♠✐♥✐♠❛❧ χ(U ) ❢♦r ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r U ✳ ❚❤❡ r❡❛♣✐♥❣ ♥✉♠❜❡r r ✐s t❤❡ ♠✐♥✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ♣s❡✉❞♦❜❛s❡ ❢♦r ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r U ✳ ❚❤❡♦r❡♠✱ ●♦❧❞st❡r♥✱ ❙❤❡❧❛❤✱ ✶✾✾✵ r < u ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈✳ ❇✉t t❤❡♥ d ≤ u ❜② ❛ t❤❡♦r❡♠ ♦❢ ❆✉❜r❡②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❉♦♠✐♥❛t✐♥❣ ♥✉♠❜❡rs✱ d

❉❡✜♥✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ ❡✈❡♥t✉❛❧ ❞♦♠✐♥❛t✐♦♥✿ ❢ ≤∗ ❣ ✐✛ ❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥② ♥✱ ❢ (♥) ≤ ❣(♥)✳ ❋♦r ❛ ✜❧t❡r F✱ ✇❡ ❞❡✜♥❡ t❤❡ r❡❞✉❝❡❞ ♦r❞❡r ❢ ≤F ❣ ✐✛ {♥ : ❢ (♥) ≤ ❣(♥)} ∈ F✳ ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② ❉ ✐s ❞♦♠✐♥❛t✐♥❣ ❬ ✲❞♦♠✐♥❛t✐♥❣❪ ✐✛ ❢♦r ❡✈❡r② ❢ t❤❡r❡ ✐s s♦♠❡ ❣ ❉ s✉❝❤ t❤❛t ❢ ❣ ❬❢ ❣❪✳ ❚❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r ❬t❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r ♦❢ ✱ ✱❪ ✐s t❤❡ s♠❛❧❧❡st ❝❛r❞✐♥❛❧ ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ ❬ ✲❞♦♠✐♥❛t✐♥❣❪ ❢❛♠✐❧② ❉ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❉♦♠✐♥❛t✐♥❣ ♥✉♠❜❡rs✱ d

❉❡✜♥✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ ❡✈❡♥t✉❛❧ ❞♦♠✐♥❛t✐♦♥✿ ❢ ≤∗ ❣ ✐✛ ❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥② ♥✱ ❢ (♥) ≤ ❣(♥)✳ ❋♦r ❛ ✜❧t❡r F✱ ✇❡ ❞❡✜♥❡ t❤❡ r❡❞✉❝❡❞ ♦r❞❡r ❢ ≤F ❣ ✐✛ {♥ : ❢ (♥) ≤ ❣(♥)} ∈ F✳ ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② ❉ ✐s ❞♦♠✐♥❛t✐♥❣ ❬F✲❞♦♠✐♥❛t✐♥❣❪ ✐✛ ❢♦r ❡✈❡r② ❢ ∈ ωω t❤❡r❡ ✐s s♦♠❡ ❣ ∈ ❉ s✉❝❤ t❤❛t ❢ ≤∗ ❣ ❬❢ ≤F ❣❪✳ ❚❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r ❬t❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r ♦❢ ✱ ✱❪ ✐s t❤❡ s♠❛❧❧❡st ❝❛r❞✐♥❛❧ ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ ❬ ✲❞♦♠✐♥❛t✐♥❣❪ ❢❛♠✐❧② ❉ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❉♦♠✐♥❛t✐♥❣ ♥✉♠❜❡rs✱ d

❉❡✜♥✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ ❡✈❡♥t✉❛❧ ❞♦♠✐♥❛t✐♦♥✿ ❢ ≤∗ ❣ ✐✛ ❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥② ♥✱ ❢ (♥) ≤ ❣(♥)✳ ❋♦r ❛ ✜❧t❡r F✱ ✇❡ ❞❡✜♥❡ t❤❡ r❡❞✉❝❡❞ ♦r❞❡r ❢ ≤F ❣ ✐✛ {♥ : ❢ (♥) ≤ ❣(♥)} ∈ F✳ ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② ❉ ✐s ❞♦♠✐♥❛t✐♥❣ ❬F✲❞♦♠✐♥❛t✐♥❣❪ ✐✛ ❢♦r ❡✈❡r② ❢ ∈ ωω t❤❡r❡ ✐s s♦♠❡ ❣ ∈ ❉ s✉❝❤ t❤❛t ❢ ≤∗ ❣ ❬❢ ≤F ❣❪✳ ❚❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r d ❬t❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r ♦❢ F✱ d(F)✱❪ ✐s t❤❡ s♠❛❧❧❡st ❝❛r❞✐♥❛❧ ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ ❬F✲❞♦♠✐♥❛t✐♥❣❪ ❢❛♠✐❧② ❉ ⊆ ωω✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❉♦♠✐♥❛t✐♥❣ ♥✉♠❜❡rs✱ d

❉❡✜♥✐t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ ❡✈❡♥t✉❛❧ ❞♦♠✐♥❛t✐♦♥✿ ❢ ≤∗ ❣ ✐✛ ❢♦r ❛❧❧ ❜✉t ✜♥✐t❡❧② ♠❛♥② ♥✱ ❢ (♥) ≤ ❣(♥)✳ ❋♦r ❛ ✜❧t❡r F✱ ✇❡ ❞❡✜♥❡ t❤❡ r❡❞✉❝❡❞ ♦r❞❡r ❢ ≤F ❣ ✐✛ {♥ : ❢ (♥) ≤ ❣(♥)} ∈ F✳ ❉❡✜♥✐t✐♦♥ ❆ ❢❛♠✐❧② ❉ ✐s ❞♦♠✐♥❛t✐♥❣ ❬F✲❞♦♠✐♥❛t✐♥❣❪ ✐✛ ❢♦r ❡✈❡r② ❢ ∈ ωω t❤❡r❡ ✐s s♦♠❡ ❣ ∈ ❉ s✉❝❤ t❤❛t ❢ ≤∗ ❣ ❬❢ ≤F ❣❪✳ ❚❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r d ❬t❤❡ ❞♦♠✐♥❛t✐♥❣ ♥✉♠❜❡r ♦❢ F✱ d(F)✱❪ ✐s t❤❡ s♠❛❧❧❡st ❝❛r❞✐♥❛❧ ♦❢ ❛ ❞♦♠✐♥❛t✐♥❣ ❬F✲❞♦♠✐♥❛t✐♥❣❪ ❢❛♠✐❧② ❉ ⊆ ωω✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ r♦❧❡ ♦❢ u ❛♥❞ d

u ❝♦♠❡s ✐♥ ❛s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ st❡♣s ✐♥ ❝♦♥str✉❝t✐♥❣ ♦♥❡ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ ♦♥❡ ❝❧❛ss✳ Pr♦♣♦s✐t✐♦♥✳ ❇❧❛ss✱ ✶✾✽✼ ❚❤❡r❡ ✐s ❛ s❡t ❉✱ ❛ s♦✲❝❛❧❧❡❞ t❡st s❡t✱ ♦❢ s✐③❡ d s✉❝❤ t❤❛t ❛♥② t✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ t❤❡r❡ ✐s s♦♠❡ ❢ ∈ ❉ ✇✐t❤ ❢ (U ) = ❢ (V )✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t✇♦ ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ✇✐t❤ st❡♣s ❛♥❞ t❛s❦s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ r♦❧❡ ♦❢ u ❛♥❞ d

u ❝♦♠❡s ✐♥ ❛s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ st❡♣s ✐♥ ❝♦♥str✉❝t✐♥❣ ♦♥❡ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ ♦♥❡ ❝❧❛ss✳ Pr♦♣♦s✐t✐♦♥✳ ❇❧❛ss✱ ✶✾✽✼ ❚❤❡r❡ ✐s ❛ s❡t ❉✱ ❛ s♦✲❝❛❧❧❡❞ t❡st s❡t✱ ♦❢ s✐③❡ d s✉❝❤ t❤❛t ❛♥② t✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ t❤❡r❡ ✐s s♦♠❡ ❢ ∈ ❉ ✇✐t❤ ❢ (U ) = ❢ (V )✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t✇♦ ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ✇✐t❤ u st❡♣s ❛♥❞ d t❛s❦s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ r♦❧❡ ♦❢ u ❛♥❞ d

u ❝♦♠❡s ✐♥ ❛s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ st❡♣s ✐♥ ❝♦♥str✉❝t✐♥❣ ♦♥❡ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ ♦♥❡ ❝❧❛ss✳ Pr♦♣♦s✐t✐♦♥✳ ❇❧❛ss✱ ✶✾✽✼ ❚❤❡r❡ ✐s ❛ s❡t ❉✱ ❛ s♦✲❝❛❧❧❡❞ t❡st s❡t✱ ♦❢ s✐③❡ d s✉❝❤ t❤❛t ❛♥② t✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ t❤❡r❡ ✐s s♦♠❡ ❢ ∈ ❉ ✇✐t❤ ❢ (U ) = ❢ (V )✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t✇♦ ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ✇✐t❤ u st❡♣s ❛♥❞ d t❛s❦s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❈❛♥❞✐❞❛t❡s

▼♦❞❡❧s ♦❢ u < d✿ ❚❤❡ ❦♥♦✇♥ ♠♦❞❡❧s ✇✐t❤ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥s ❢✉❧✜❧ t❤❡ str♦♥❣❡r ✐♥❡q✉❛❧✐t② u < g✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ◆❈❋✳ ❚❤❡r❡ ✐s ♦♥❡ t②♣❡ ♦❢ ♠♦❞❡❧ ✭❢r♦♠ ❬❇s❙❤✿✷✺✼❪✱ ✶✾✽✾✮ ♦❢ u < d ❣♦tt❡♥ ✇✐t❤ ❛ ✜♥✐t❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ❝✳❝✳❝✳ ♣❛rt✐❛❧ ♦r❞❡rs✳ ❚❤❡♦r❡♠✳ ▼✳ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs ❛♥❞ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❈❛♥❞✐❞❛t❡s

▼♦❞❡❧s ♦❢ u < d✿ ❚❤❡ ❦♥♦✇♥ ♠♦❞❡❧s ✇✐t❤ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥s ❢✉❧✜❧ t❤❡ str♦♥❣❡r ✐♥❡q✉❛❧✐t② u < g✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ◆❈❋✳ ❚❤❡r❡ ✐s ♦♥❡ t②♣❡ ♦❢ ♠♦❞❡❧ ✭❢r♦♠ ❬❇s❙❤✿✷✺✼❪✱ ✶✾✽✾✮ ♦❢ u < d ❣♦tt❡♥ ✇✐t❤ ❛ ✜♥✐t❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ❝✳❝✳❝✳ ♣❛rt✐❛❧ ♦r❞❡rs✳ ❚❤❡♦r❡♠✳ ▼✳ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs ❛♥❞ u < d✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❈❛♥❞✐❞❛t❡s

▼♦❞❡❧s ♦❢ u < d✿ ❚❤❡ ❦♥♦✇♥ ♠♦❞❡❧s ✇✐t❤ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥s ❢✉❧✜❧ t❤❡ str♦♥❣❡r ✐♥❡q✉❛❧✐t② u < g✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ◆❈❋✳ ❚❤❡r❡ ✐s ♦♥❡ t②♣❡ ♦❢ ♠♦❞❡❧ ✭❢r♦♠ ❬❇s❙❤✿✷✺✼❪✱ ✶✾✽✾✮ ♦❢ u < d ❣♦tt❡♥ ✇✐t❤ ❛ ✜♥✐t❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ♦❢ ❝✳❝✳❝✳ ♣❛rt✐❛❧ ♦r❞❡rs✳ ❚❤❡♦r❡♠✳ ▼✳ ■t ✐s ❝♦♥s✐st❡♥t r❡❧❛t✐✈❡ t♦ ❩❋❈ t❤❛t t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs ❛♥❞ u < d✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ

✻

▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ

✻

▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✻ ✻ ✻

❱ r s ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ r s

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✻ ✻ ✻ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻

ν ❱ [(rα : α < δ)][(sξ : ξ < ν)]) ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ [(rα : α < δ)][(sξ : ξ < ν)])

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆♥ ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ r❡❝t❛♥❣✉❧❛r str✉❝t✉r❡

✲

ν ❈♦❤❡♥ r❡❛❧s rα δ ▼❛t❤✐❛s r❡❛❧s sξ

✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✲ ✻ ✻ ✻ ✻ ✻ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻

ν ❱ [(rα : α < δ)][(sξ : ξ < ν)]) ❋✐❣✉r❡ ✶✿ ❆ s❦❡t❝❤ ♦❢ ❱ [(rα : α < δ)][(sξ : ξ < ν)])

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♣❧✐tt✐♥❣ ❢❛♠✐❧✐❡s ❛♥❞ r❡❛♣✐♥❣

❉❡✜♥✐t✐♦♥ S ⊆ [ω]ω ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✐✛ (∀❳ ∈ [ω]ω)(∃❙ ∈ S )(❳ ∩ ❙ ❛♥❞ ❳ ❙ ❛r❡ ❜♦t❤ ✐♥✜♥✐t❡✮✳ ❚❤❡ s♣❧✐tt✐♥❣ ♥✉♠❜❡r s ✐s t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧②✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ▼✳✱ ✶✾✾✾ ■❢ t❤❡♥ t❤❡r❡ ❛r❡ ❛t ♠♦st t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❆✉❜r❡②✱ ✷✵✵✹ ■❢ t❤❡♥ ✳ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ ✱ ✇❡ ❤❛✈❡ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♣❧✐tt✐♥❣ ❢❛♠✐❧✐❡s ❛♥❞ r❡❛♣✐♥❣

❉❡✜♥✐t✐♦♥ S ⊆ [ω]ω ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✐✛ (∀❳ ∈ [ω]ω)(∃❙ ∈ S )(❳ ∩ ❙ ❛♥❞ ❳ ❙ ❛r❡ ❜♦t❤ ✐♥✜♥✐t❡✮✳ ❚❤❡ s♣❧✐tt✐♥❣ ♥✉♠❜❡r s ✐s t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧②✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ▼✳✱ ✶✾✾✾ ■❢ s > r t❤❡♥ t❤❡r❡ ❛r❡ ❛t ♠♦st t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❆✉❜r❡②✱ ✷✵✵✹ ■❢ t❤❡♥ ✳ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ ✱ ✇❡ ❤❛✈❡ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♣❧✐tt✐♥❣ ❢❛♠✐❧✐❡s ❛♥❞ r❡❛♣✐♥❣

❉❡✜♥✐t✐♦♥ S ⊆ [ω]ω ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✐✛ (∀❳ ∈ [ω]ω)(∃❙ ∈ S )(❳ ∩ ❙ ❛♥❞ ❳ ❙ ❛r❡ ❜♦t❤ ✐♥✜♥✐t❡✮✳ ❚❤❡ s♣❧✐tt✐♥❣ ♥✉♠❜❡r s ✐s t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧②✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ▼✳✱ ✶✾✾✾ ■❢ s > r t❤❡♥ t❤❡r❡ ❛r❡ ❛t ♠♦st t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❆✉❜r❡②✱ ✷✵✵✹ ■❢ r < d t❤❡♥ r = u✳ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ ✱ ✇❡ ❤❛✈❡ ✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♣❧✐tt✐♥❣ ❢❛♠✐❧✐❡s ❛♥❞ r❡❛♣✐♥❣

❉❡✜♥✐t✐♦♥ S ⊆ [ω]ω ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✐✛ (∀❳ ∈ [ω]ω)(∃❙ ∈ S )(❳ ∩ ❙ ❛♥❞ ❳ ❙ ❛r❡ ❜♦t❤ ✐♥✜♥✐t❡✮✳ ❚❤❡ s♣❧✐tt✐♥❣ ♥✉♠❜❡r s ✐s t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧②✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ▼✳✱ ✶✾✾✾ ■❢ s > r t❤❡♥ t❤❡r❡ ❛r❡ ❛t ♠♦st t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❆✉❜r❡②✱ ✷✵✵✹ ■❢ r < d t❤❡♥ r = u✳ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ P✱ ✇❡ ❤❛✈❡ ν = u = r✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♣❧✐tt✐♥❣ ❢❛♠✐❧✐❡s ❛♥❞ r❡❛♣✐♥❣

❉❡✜♥✐t✐♦♥ S ⊆ [ω]ω ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✐✛ (∀❳ ∈ [ω]ω)(∃❙ ∈ S )(❳ ∩ ❙ ❛♥❞ ❳ ❙ ❛r❡ ❜♦t❤ ✐♥✜♥✐t❡✮✳ ❚❤❡ s♣❧✐tt✐♥❣ ♥✉♠❜❡r s ✐s t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧②✳ ❚❤❡♦r❡♠✳ ❇❧❛ss✱ ▼✳✱ ✶✾✾✾ ■❢ s > r t❤❡♥ t❤❡r❡ ❛r❡ ❛t ♠♦st t✇♦ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳ ❚❤❡♦r❡♠✳ ❆✉❜r❡②✱ ✷✵✵✹ ■❢ r < d t❤❡♥ r = u✳ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ P✱ ✇❡ ❤❛✈❡ ν = u = r✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ s♠❛❧❧ s♣❧✐tt✐♥❣ ❢❛♠✐❧②

Pr♦♣♦s✐t✐♦♥ ■♥ ❱ P✱ s ≤ ν✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❘❡♠❡♠❜❡r✱ sξ✱ ξ < ν✱ ❛r❡ t❤❡ ▼❛t❤✐❛s r❡❛❧s✳ ❲❡ s❡t ❳ξ = {♥ ∈ ω : |sξ ∩ ♥| ✐s ❡✈❡♥}. ❚❤❡♥ ❳ ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✇✐t♥❡ss✐♥❣ ✳ ✭❲❡ ❤❛✈❡ ✐♥ t❤❡s❡ ♠♦❞❡❧s✳✮ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ ✱ ✇❡ ❤❛✈❡ ❛♥❞ ❤❡♥❝❡ t❤❡ ✲❚❤❡♦r❡♠ ❞♦❡s ♥♦t ❛♣♣❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ s♠❛❧❧ s♣❧✐tt✐♥❣ ❢❛♠✐❧②

Pr♦♣♦s✐t✐♦♥ ■♥ ❱ P✱ s ≤ ν✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❘❡♠❡♠❜❡r✱ sξ✱ ξ < ν✱ ❛r❡ t❤❡ ▼❛t❤✐❛s r❡❛❧s✳ ❲❡ s❡t ❳ξ = {♥ ∈ ω : |sξ ∩ ♥| ✐s ❡✈❡♥}. ❚❤❡♥ {❳ξ : ξ < ν} ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✇✐t♥❡ss✐♥❣ s ≤ r✳

• ✭❲❡ ❤❛✈❡

✐♥ t❤❡s❡ ♠♦❞❡❧s✳✮ ❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ ✱ ✇❡ ❤❛✈❡ ❛♥❞ ❤❡♥❝❡ t❤❡ ✲❚❤❡♦r❡♠ ❞♦❡s ♥♦t ❛♣♣❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ s♠❛❧❧ s♣❧✐tt✐♥❣ ❢❛♠✐❧②

Pr♦♣♦s✐t✐♦♥ ■♥ ❱ P✱ s ≤ ν✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❘❡♠❡♠❜❡r✱ sξ✱ ξ < ν✱ ❛r❡ t❤❡ ▼❛t❤✐❛s r❡❛❧s✳ ❲❡ s❡t ❳ξ = {♥ ∈ ω : |sξ ∩ ♥| ✐s ❡✈❡♥}. ❚❤❡♥ {❳ξ : ξ < ν} ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✇✐t♥❡ss✐♥❣ s ≤ r✳

• ✭❲❡ ❤❛✈❡ s = ν ✐♥ t❤❡s❡ ♠♦❞❡❧s✳✮

❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ ✱ ✇❡ ❤❛✈❡ ❛♥❞ ❤❡♥❝❡ t❤❡ ✲❚❤❡♦r❡♠ ❞♦❡s ♥♦t ❛♣♣❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ s♠❛❧❧ s♣❧✐tt✐♥❣ ❢❛♠✐❧②

Pr♦♣♦s✐t✐♦♥ ■♥ ❱ P✱ s ≤ ν✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❘❡♠❡♠❜❡r✱ sξ✱ ξ < ν✱ ❛r❡ t❤❡ ▼❛t❤✐❛s r❡❛❧s✳ ❲❡ s❡t ❳ξ = {♥ ∈ ω : |sξ ∩ ♥| ✐s ❡✈❡♥}. ❚❤❡♥ {❳ξ : ξ < ν} ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✇✐t♥❡ss✐♥❣ s ≤ r✳

• ✭❲❡ ❤❛✈❡ s = ν ✐♥ t❤❡s❡ ♠♦❞❡❧s✳✮

❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ P✱ ✇❡ ❤❛✈❡ s ≤ ν = u = r ❛♥❞ ❤❡♥❝❡ t❤❡ r < s✲❚❤❡♦r❡♠ ❞♦❡s ♥♦t ❛♣♣❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ s♠❛❧❧ s♣❧✐tt✐♥❣ ❢❛♠✐❧②

Pr♦♣♦s✐t✐♦♥ ■♥ ❱ P✱ s ≤ ν✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❘❡♠❡♠❜❡r✱ sξ✱ ξ < ν✱ ❛r❡ t❤❡ ▼❛t❤✐❛s r❡❛❧s✳ ❲❡ s❡t ❳ξ = {♥ ∈ ω : |sξ ∩ ♥| ✐s ❡✈❡♥}. ❚❤❡♥ {❳ξ : ξ < ν} ✐s ❛ s♣❧✐tt✐♥❣ ❢❛♠✐❧② ✇✐t♥❡ss✐♥❣ s ≤ r✳

• ✭❲❡ ❤❛✈❡ s = ν ✐♥ t❤❡s❡ ♠♦❞❡❧s✳✮

❈♦♥❝❧✉s✐♦♥✿ ■♥ ❱ P✱ ✇❡ ❤❛✈❡ s ≤ ν = u = r ❛♥❞ ❤❡♥❝❡ t❤❡ r < s✲❚❤❡♦r❡♠ ❞♦❡s ♥♦t ❛♣♣❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ ❛♥s✇❡r

❈❧❛✐♠ ■♥ t❤❡ ♠♦❞❡❧ ❱ P t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❲❡ ❤❛✈❡ t❤❡ ✉❧tr❛✜❧t❡r UP t❤❛t ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ▼❛t❤✐❛s r❡❛❧s sξ✱ ξ < ν✳ ❇② ❬❇▼❪ ❛❧❧ ✉❧tr❛✜❧t❡rs ✇✐t❤ ❣❡♥❡r❛t♦rs ❤❛✈❡ ❝❢ ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t t♦

P✳

❲❡ s❤❛❧❧ s❤♦✇ t❤❛t t❤❡r❡ ✐s ❛ ✜❧t❡r

✵ t❤❛t ✐s ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t

t♦

P s✉❝❤ t❤❛t ✵ ❡①t❡♥❞❡❞ ❜② ❢❡✇❡r t❤❛♥ ✵ s❡ts ✐s ♥♦t

❛❧♠♦st ✉❧tr❛✳ ❲❡ s❤❛❧❧ ❣❡t

✵ ❢r♦♠ t❤❡ ❈♦❤❡♥ r❡❛❧s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ ❛♥s✇❡r

❈❧❛✐♠ ■♥ t❤❡ ♠♦❞❡❧ ❱ P t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❲❡ ❤❛✈❡ t❤❡ ✉❧tr❛✜❧t❡r UP t❤❛t ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ▼❛t❤✐❛s r❡❛❧s sξ✱ ξ < ν✳ ❇② ❬❇▼❪ ❛❧❧ ✉❧tr❛✜❧t❡rs U ✇✐t❤ < d ❣❡♥❡r❛t♦rs ❤❛✈❡ ❝❢(ωω/U ) = d > r ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP✳ ❲❡ s❤❛❧❧ s❤♦✇ t❤❛t t❤❡r❡ ✐s ❛ ✜❧t❡r

✵ t❤❛t ✐s ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t

t♦

P s✉❝❤ t❤❛t ✵ ❡①t❡♥❞❡❞ ❜② ❢❡✇❡r t❤❛♥ ✵ s❡ts ✐s ♥♦t

❛❧♠♦st ✉❧tr❛✳ ❲❡ s❤❛❧❧ ❣❡t

✵ ❢r♦♠ t❤❡ ❈♦❤❡♥ r❡❛❧s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ ❛♥s✇❡r

❈❧❛✐♠ ■♥ t❤❡ ♠♦❞❡❧ ❱ P t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❲❡ ❤❛✈❡ t❤❡ ✉❧tr❛✜❧t❡r UP t❤❛t ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ▼❛t❤✐❛s r❡❛❧s sξ✱ ξ < ν✳ ❇② ❬❇▼❪ ❛❧❧ ✉❧tr❛✜❧t❡rs U ✇✐t❤ < d ❣❡♥❡r❛t♦rs ❤❛✈❡ ❝❢(ωω/U ) = d > r ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP✳ ❲❡ s❤❛❧❧ s❤♦✇ t❤❛t t❤❡r❡ ✐s ❛ ✜❧t❡r H✵ t❤❛t ✐s ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t t♦ UP s✉❝❤ t❤❛t H✵ ❡①t❡♥❞❡❞ ❜② ❢❡✇❡r t❤❛♥ d(H✵) s❡ts ✐s ♥♦t ❛❧♠♦st ✉❧tr❛✳ ❲❡ s❤❛❧❧ ❣❡t

✵ ❢r♦♠ t❤❡ ❈♦❤❡♥ r❡❛❧s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ ❛♥s✇❡r

❈❧❛✐♠ ■♥ t❤❡ ♠♦❞❡❧ ❱ P t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❲❡ ❤❛✈❡ t❤❡ ✉❧tr❛✜❧t❡r UP t❤❛t ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ▼❛t❤✐❛s r❡❛❧s sξ✱ ξ < ν✳ ❇② ❬❇▼❪ ❛❧❧ ✉❧tr❛✜❧t❡rs U ✇✐t❤ < d ❣❡♥❡r❛t♦rs ❤❛✈❡ ❝❢(ωω/U ) = d > r ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP✳ ❲❡ s❤❛❧❧ s❤♦✇ t❤❛t t❤❡r❡ ✐s ❛ ✜❧t❡r H✵ t❤❛t ✐s ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t t♦ UP s✉❝❤ t❤❛t H✵ ❡①t❡♥❞❡❞ ❜② ❢❡✇❡r t❤❛♥ d(H✵) s❡ts ✐s ♥♦t ❛❧♠♦st ✉❧tr❛✳ ❲❡ s❤❛❧❧ ❣❡t H✵ ❢r♦♠ t❤❡ ❈♦❤❡♥ r❡❛❧s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❚❤❡ ❛♥s✇❡r

❈❧❛✐♠ ■♥ t❤❡ ♠♦❞❡❧ ❱ P t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✿ ❲❡ ❤❛✈❡ t❤❡ ✉❧tr❛✜❧t❡r UP t❤❛t ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ▼❛t❤✐❛s r❡❛❧s sξ✱ ξ < ν✳ ❇② ❬❇▼❪ ❛❧❧ ✉❧tr❛✜❧t❡rs U ✇✐t❤ < d ❣❡♥❡r❛t♦rs ❤❛✈❡ ❝❢(ωω/U ) = d > r ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP✳ ❲❡ s❤❛❧❧ s❤♦✇ t❤❛t t❤❡r❡ ✐s ❛ ✜❧t❡r H✵ t❤❛t ✐s ♥♦♥✲♥❡❛r❧②✲❝♦❤❡r❡♥t t♦ UP s✉❝❤ t❤❛t H✵ ❡①t❡♥❞❡❞ ❜② ❢❡✇❡r t❤❛♥ d(H✵) s❡ts ✐s ♥♦t ❛❧♠♦st ✉❧tr❛✳ ❲❡ s❤❛❧❧ ❣❡t H✵ ❢r♦♠ t❤❡ ❈♦❤❡♥ r❡❛❧s✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❋✐❧t❡rs ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP

❲❡ t❤✐♥❦ ♦❢ t❤❡ ❈♦❤❡♥ r❡❛❧s ❛s s✉❜s❡ts ♦❢ ω ❛♥❞ ❧❡t t❤❡ ❈♦❤❡♥ r❡❛❧s rα✱ α < δ✱ ❜❡ t❤❡✐r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❲❡ s❡t ❳α,ξ = {rα(♥) : |sξ ∩ ♥| ❡✈❡♥}, ❳ ❳ ✐s ❛♥ ❡✈❡♥ ♦r❞✐♥❛❧

❢✉❧❧

❳ ▲❡♠♠❛

❢✉❧❧ ❤❛s t❤❡ ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❋✐❧t❡rs ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP

❲❡ t❤✐♥❦ ♦❢ t❤❡ ❈♦❤❡♥ r❡❛❧s ❛s s✉❜s❡ts ♦❢ ω ❛♥❞ ❧❡t t❤❡ ❈♦❤❡♥ r❡❛❧s rα✱ α < δ✱ ❜❡ t❤❡✐r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❲❡ s❡t ❳α,ξ = {rα(♥) : |sξ ∩ ♥| ❡✈❡♥}, Hξ = {❳α,ξ : α < δ}, ❳ ✐s ❛♥ ❡✈❡♥ ♦r❞✐♥❛❧

❢✉❧❧

❳ ▲❡♠♠❛

❢✉❧❧ ❤❛s t❤❡ ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❋✐❧t❡rs ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP

❲❡ t❤✐♥❦ ♦❢ t❤❡ ❈♦❤❡♥ r❡❛❧s ❛s s✉❜s❡ts ♦❢ ω ❛♥❞ ❧❡t t❤❡ ❈♦❤❡♥ r❡❛❧s rα✱ α < δ✱ ❜❡ t❤❡✐r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❲❡ s❡t ❳α,ξ = {rα(♥) : |sξ ∩ ♥| ❡✈❡♥}, Hξ = {❳α,ξ : α < δ}, H = {❳α,ξ : α < δ, ξ < ν, ξ ✐s ❛♥ ❡✈❡♥ ♦r❞✐♥❛❧},

❢✉❧❧

❳ ▲❡♠♠❛

❢✉❧❧ ❤❛s t❤❡ ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❋✐❧t❡rs ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP

❲❡ t❤✐♥❦ ♦❢ t❤❡ ❈♦❤❡♥ r❡❛❧s ❛s s✉❜s❡ts ♦❢ ω ❛♥❞ ❧❡t t❤❡ ❈♦❤❡♥ r❡❛❧s rα✱ α < δ✱ ❜❡ t❤❡✐r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❲❡ s❡t ❳α,ξ = {rα(♥) : |sξ ∩ ♥| ❡✈❡♥}, Hξ = {❳α,ξ : α < δ}, H = {❳α,ξ : α < δ, ξ < ν, ξ ✐s ❛♥ ❡✈❡♥ ♦r❞✐♥❛❧}, H❢✉❧❧ = {❳α,ξ : α < δ, ξ < ν}. ▲❡♠♠❛

❢✉❧❧ ❤❛s t❤❡ ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❋✐❧t❡rs ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP

❲❡ t❤✐♥❦ ♦❢ t❤❡ ❈♦❤❡♥ r❡❛❧s ❛s s✉❜s❡ts ♦❢ ω ❛♥❞ ❧❡t t❤❡ ❈♦❤❡♥ r❡❛❧s rα✱ α < δ✱ ❜❡ t❤❡✐r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❲❡ s❡t ❳α,ξ = {rα(♥) : |sξ ∩ ♥| ❡✈❡♥}, Hξ = {❳α,ξ : α < δ}, H = {❳α,ξ : α < δ, ξ < ν, ξ ✐s ❛♥ ❡✈❡♥ ♦r❞✐♥❛❧}, H❢✉❧❧ = {❳α,ξ : α < δ, ξ < ν}. ▲❡♠♠❛ H❢✉❧❧ ❤❛s t❤❡ ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❋✐❧t❡rs ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ UP

❲❡ t❤✐♥❦ ♦❢ t❤❡ ❈♦❤❡♥ r❡❛❧s ❛s s✉❜s❡ts ♦❢ ω ❛♥❞ ❧❡t t❤❡ ❈♦❤❡♥ r❡❛❧s rα✱ α < δ✱ ❜❡ t❤❡✐r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❲❡ s❡t ❳α,ξ = {rα(♥) : |sξ ∩ ♥| ❡✈❡♥}, Hξ = {❳α,ξ : α < δ}, H = {❳α,ξ : α < δ, ξ < ν, ξ ✐s ❛♥ ❡✈❡♥ ♦r❞✐♥❛❧}, H❢✉❧❧ = {❳α,ξ : α < δ, ξ < ν}. ▲❡♠♠❛ H❢✉❧❧ ❤❛s t❤❡ ✜♥✐t❡ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❢ (H✵) ⊆ ❢ (UP)

▲❡♠♠❛ ❋♦r ❡✈❡r② ξ < ν✱ ❢♦r ❡✈❡r② ❨ ∈ ❱ (δ, ξ) ❢♦r ❡✈❡r② α✐ < δ✱ ✐ < ❦✱ ✇❡ ❤❛✈❡✿ ■❢ ❨ ∩

✐<❦ r❛♥❣❡(rα✐) ✐s ✐♥✜♥✐t❡✱ t❤❡♥ t❤❡ s❡t

❨ ∩

• ✵≤✐<❦

❳α✐,ξ ✐s ✐♥✜♥✐t❡✳ ▲❡♠♠❛ ❋♦r ❡✈❡r② ✜♥✐t❡✲t♦✲♦♥❡ ❢ ✱ ❢ (H✵) ⊆ ❢ (UP)✳ ❙♦

✵ ❛♥❞ P ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ❛♥❞ t❤✉s ✇❡ ❤❛✈❡ ❛t ❧❡❛st

t✇♦ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❢ (H✵) ⊆ ❢ (UP)

▲❡♠♠❛ ❋♦r ❡✈❡r② ξ < ν✱ ❢♦r ❡✈❡r② ❨ ∈ ❱ (δ, ξ) ❢♦r ❡✈❡r② α✐ < δ✱ ✐ < ❦✱ ✇❡ ❤❛✈❡✿ ■❢ ❨ ∩

✐<❦ r❛♥❣❡(rα✐) ✐s ✐♥✜♥✐t❡✱ t❤❡♥ t❤❡ s❡t

❨ ∩

• ✵≤✐<❦

❳α✐,ξ ✐s ✐♥✜♥✐t❡✳ ▲❡♠♠❛ ❋♦r ❡✈❡r② ✜♥✐t❡✲t♦✲♦♥❡ ❢ ✱ ❢ (H✵) ⊆ ❢ (UP)✳ ❙♦ H✵ ❛♥❞ UP ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ❛♥❞ t❤✉s ✇❡ ❤❛✈❡ ❛t ❧❡❛st t✇♦ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❢ (H✵) ⊆ ❢ (UP)

▲❡♠♠❛ ❋♦r ❡✈❡r② ξ < ν✱ ❢♦r ❡✈❡r② ❨ ∈ ❱ (δ, ξ) ❢♦r ❡✈❡r② α✐ < δ✱ ✐ < ❦✱ ✇❡ ❤❛✈❡✿ ■❢ ❨ ∩

✐<❦ r❛♥❣❡(rα✐) ✐s ✐♥✜♥✐t❡✱ t❤❡♥ t❤❡ s❡t

❨ ∩

• ✵≤✐<❦

❳α✐,ξ ✐s ✐♥✜♥✐t❡✳ ▲❡♠♠❛ ❋♦r ❡✈❡r② ✜♥✐t❡✲t♦✲♦♥❡ ❢ ✱ ❢ (H✵) ⊆ ❢ (UP)✳ ❙♦ H✵ ❛♥❞ UP ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ❛♥❞ t❤✉s ✇❡ ❤❛✈❡ ❛t ❧❡❛st t✇♦ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♠❛❧❧ ❞♦♠✐♥❛t✐♥❣ ❢❛♠✐❧✐❡s ♠♦❞✉❧♦ ✜❧t❡r ♦r❞❡r✐♥❣s

❆✐♠✿ ❋✐♥❞ ❛ tr❡❡ ♦❢ ♣❛✐r✇✐s❡ ♥♦♥✲♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❛♠♦♥❣ t❤❡ s✉♣❡rs❡ts ♦❢ H✵✳ Pr♦♣♦s✐t✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ■❢ ❛ ✜❧t❡r F ❛♥❞ ❛ ✉❧tr❛✜❧t❡r U ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ t❤❡♥ d(F) ≤ χ(U )✳ ❙♦ ✐♥ ❱ ✱

✵

✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♠❛❧❧ ❞♦♠✐♥❛t✐♥❣ ❢❛♠✐❧✐❡s ♠♦❞✉❧♦ ✜❧t❡r ♦r❞❡r✐♥❣s

❆✐♠✿ ❋✐♥❞ ❛ tr❡❡ ♦❢ ♣❛✐r✇✐s❡ ♥♦♥✲♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❛♠♦♥❣ t❤❡ s✉♣❡rs❡ts ♦❢ H✵✳ Pr♦♣♦s✐t✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ■❢ ❛ ✜❧t❡r F ❛♥❞ ❛ ✉❧tr❛✜❧t❡r U ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ t❤❡♥ d(F) ≤ χ(U )✳ ❙♦ ✐♥ ❱ P✱ d(H✵) ≤ ν✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♠❛❧❧ ❞♦♠✐♥❛t✐♥❣ ❢❛♠✐❧✐❡s ♠♦❞✉❧♦ ✜❧t❡r ♦r❞❡r✐♥❣s

❆✐♠✿ ❋✐♥❞ ❛ tr❡❡ ♦❢ ♣❛✐r✇✐s❡ ♥♦♥✲♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❛♠♦♥❣ t❤❡ s✉♣❡rs❡ts ♦❢ H✵✳ Pr♦♣♦s✐t✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ ■❢ ❛ ✜❧t❡r F ❛♥❞ ❛ ✉❧tr❛✜❧t❡r U ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ t❤❡♥ d(F) ≤ χ(U )✳ ❙♦ ✐♥ ❱ P✱ d(H✵) ≤ ν✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♠❛❧❧ t❡st s❡ts

▲❡t t(F) ❜❡ t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ t❡st s❡t ❢♦r ♥❡❛r ❝♦❤❡r❡♥❝❡ ✐♥ [F] = {G : G ✜❧t❡r, G ⊇ F}✳ Pr♦♣♦s✐t✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ t(F) ≤ d(F)✳ ❙♦✱ ✐♥ ❱ ✱ t

✵

✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♠❛❧❧ t❡st s❡ts

▲❡t t(F) ❜❡ t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ t❡st s❡t ❢♦r ♥❡❛r ❝♦❤❡r❡♥❝❡ ✐♥ [F] = {G : G ✜❧t❡r, G ⊇ F}✳ Pr♦♣♦s✐t✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ t(F) ≤ d(F)✳ ❙♦✱ ✐♥ ❱ P✱ t(H✵) ≤ ν✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❙♠❛❧❧ t❡st s❡ts

▲❡t t(F) ❜❡ t❤❡ s♠❛❧❧❡st s✐③❡ ♦❢ ❛ t❡st s❡t ❢♦r ♥❡❛r ❝♦❤❡r❡♥❝❡ ✐♥ [F] = {G : G ✜❧t❡r, G ⊇ F}✳ Pr♦♣♦s✐t✐♦♥✳ ❇❛♥❛❦❤✱ ❇❧❛ss✱ ✷✵✵✺ t(F) ≤ d(F)✳ ❙♦✱ ✐♥ ❱ P✱ t(H✵) ≤ ν✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ tr❡❡ ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

▲❡♠♠❛✳ ❙❧✐❣❤t ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❇❧❛ss✱ ✶✾✽✼ ■❢ ❛❧❧ ❡①t❡♥s✐♦♥s ♦❢ H✵ ❜② ❢❡✇❡r t❤❛♥ t(H✵) s❡ts ❛r❡ ♥♦t ❛❧♠♦st ✉❧tr❛✱ t❤❡♥ ✇❡ ❝❛♥ ❝♦♥str✉❝t ✐♥✜♥✐t❡❧② ♠❛♥② ♣❛✐r✇✐s❡ ♥♦♥✲♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❜② ❛♥ ✐♥❞✉❝t✐♦♥ ♦❢ ❧❡♥❣t❤ t(H✵)✳ ❙♦ ♦✉r ♣r♦♦❢ ✐s ✜♥✐s❤❡❞ ✇✐t❤ ▲❡♠♠❛ ■♥ ❱ P✱ ❡❛❝❤ ❡①t❡♥s✐♦♥ ♦❢ H✵ ❜② ❢❡✇❡r t❤❛♥ ν s❡ts ✐s ♥♦t ❛❧♠♦st ✉❧tr❛✳ ❛♥❞ ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ ❝♦♥str✉❝t t❤❡ ❞❡s✐r❡❞ tr❡❡ ❡①♣❧✐❝✐t❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❆ tr❡❡ ♦❢ ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s

▲❡♠♠❛✳ ❙❧✐❣❤t ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❇❧❛ss✱ ✶✾✽✼ ■❢ ❛❧❧ ❡①t❡♥s✐♦♥s ♦❢ H✵ ❜② ❢❡✇❡r t❤❛♥ t(H✵) s❡ts ❛r❡ ♥♦t ❛❧♠♦st ✉❧tr❛✱ t❤❡♥ ✇❡ ❝❛♥ ❝♦♥str✉❝t ✐♥✜♥✐t❡❧② ♠❛♥② ♣❛✐r✇✐s❡ ♥♦♥✲♥❡❛r❧② ❝♦❤❡r❡♥t ✉❧tr❛✜❧t❡rs ❜② ❛♥ ✐♥❞✉❝t✐♦♥ ♦❢ ❧❡♥❣t❤ t(H✵)✳ ❙♦ ♦✉r ♣r♦♦❢ ✐s ✜♥✐s❤❡❞ ✇✐t❤ ▲❡♠♠❛ ■♥ ❱ P✱ ❡❛❝❤ ❡①t❡♥s✐♦♥ ♦❢ H✵ ❜② ❢❡✇❡r t❤❛♥ ν s❡ts ✐s ♥♦t ❛❧♠♦st ✉❧tr❛✳ ❛♥❞ ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ ❝♦♥str✉❝t t❤❡ ❞❡s✐r❡❞ tr❡❡ ❡①♣❧✐❝✐t❧②✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇✐❜❧✐♦❣r❛♣❤②✱ ♣❛rt ✶

❏❛s♦♥ ❆✉❜r❡②✳ ❈♦♠❜✐♥❛t♦r✐❝s ❢♦r t❤❡ ❉♦♠✐♥❛t✐♥❣ ❛♥❞ t❤❡ ❯♥s♣❧✐tt✐♥❣ ◆✉♠❜❡rs✳ ❏✳ ❙②♠❜♦❧✐❝ ▲♦❣✐❝✱ ✻✾✿✹✽✷✕✹✾✽✱ ✷✵✵✹✳ ❚❛r❛s ❇❛♥❛❦❤ ❛♥❞ ❆♥❞r❡❛s ❇❧❛ss✳ ❚❤❡ ◆✉♠❜❡r ♦❢ ◆❡❛r✲❈♦❤❡r❡♥❝❡ ❈❧❛ss❡s ♦❢ ❯❧tr❛✜❧t❡rs ✐s ❊✐t❤❡r ❋✐♥✐t❡ ♦r ✷c✳ ❙❡t ❚❤❡♦r② ✭Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❙♣❡❝✐❛❧ ❨❡❛r ✐♥ ❙❡t ❚❤❡♦r② ✐♥ ❇❛r❝❡❧♦♥❛✱ ✷✵✵✸✴✷✵✵✹✮✳ P❛❣❡s ✷✺✼ ✕ ✷✼✸✳ ❊❞s✳ ❏✳ ❇❛❣❛r✐❛ ❛♥❞ ❙✳ ❚♦❞♦r❝❡✈✐❝✳ ❙❡r✐❡s✿ ❚r❡♥❞s ✐♥ ▼❛t❤❡♠❛t✐❝s✳ ❇✐r❦❤❛❡✉s❡r✱ ✷✵✵✻✳ ❆♥❞r❡❛s ❇❧❛ss✳ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs✱ ■■✿ ❆♣♣❧✐❝❛t✐♦♥s t♦ ♦♣❡r❛t♦r ✐❞❡❛❧s✱ t❤❡ ❙t♦♥❡✲❷❡❝❤ r❡♠❛✐♥❞❡r ♦❢ ❛ ❤❛❧❢✲❧✐♥❡✱ ♦r❞❡r ✐❞❡❛❧s ♦❢ s❡q✉❡♥❝❡s✱ ❛♥❞ s❧❡♥❞❡r♥❡ss ♦❢ ❣r♦✉♣s✳ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✵✵✿✺✺✼✕✺✽✶✱ ✶✾✽✼✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇✐❜❧✐♦❣r❛♣❤②✱ ♣❛rt ✷

❆♥❞r❡❛s ❇❧❛ss✳ ●r♦✉♣✇✐s❡ ❞❡♥s✐t② ❛♥❞ r❡❧❛t❡❞ ❝❛r❞✐♥❛❧s✳ ❆r❝❤✳ ▼❛t❤✳ ▲♦❣✐❝✱ ✸✵✿✶✕✶✶✱ ✶✾✾✵✳ ❆♥❞r❡❛s ❇❧❛ss ❛♥❞ ❈❧❛✉❞❡ ▲❛✢❛♠♠❡✳ ❈♦♥s✐st❡♥❝② r❡s✉❧ts ❛❜♦✉t ✜❧t❡rs ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐♥❡q✉✐✈❛❧❡♥t ❣r♦✇t❤ t②♣❡s✳ ❏✳ ❙②♠❜♦❧✐❝ ▲♦❣✐❝✱ ✺✹✿✺✵✕✺✻✱ ✶✾✽✾✳ ❆♥❞r❡❛s ❇❧❛ss ❛♥❞ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r✳ ❖♥ t❤❡ ❝♦✜♥❛❧✐t② ♦❢ ✉❧tr❛♣♦✇❡rs✳ ❏✳ ❙②♠❜♦❧✐❝ ▲♦❣✐❝✱ ✻✹✿✼✷✼✕✼✸✻✱ ✶✾✾✾✳ ❆♥❞r❡❛s ❇❧❛ss ❛♥❞ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✳ ❬❇s❙❤✿✷✹✷❪✳ ❚❤❡r❡ ♠❛② ❜❡ s✐♠♣❧❡ Pℵ✶✲ ❛♥❞ Pℵ✷✲♣♦✐♥ts ❛♥❞ t❤❡ ❘✉❞✐♥✲❑❡✐s❧❡r ♦r❞❡r✐♥❣ ♠❛② ❜❡ ❞♦✇♥✇❛r❞ ❞✐r❡❝t❡❞✳ ❆♥♥✳ P✉r❡ ❆♣♣❧✳ ▲♦❣✐❝✱ ✸✸✿✷✶✸✕✷✹✸✱ ❬❇s❙❤✿✷✹✷❪✱ ✶✾✽✼✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇✐❜❧✐♦❣r❛♣❤②✱ ♣❛rt ✸

❆♥❞r❡❛s ❇❧❛ss ❛♥❞ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✳ ❬❇s❙❤✿✷✽✼❪✳ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ■■■✿ ❆ s✐♠♣❧✐✜❡❞ ❝♦♥s✐st❡♥❝② ♣r♦♦❢✳ ◆♦tr❡ ❉❛♠❡ ❏✳ ❋♦r♠❛❧ ▲♦❣✐❝✱ ✸✵✿✺✸✵✕✺✸✽✱ ❬❇s❙❤✿✷✽✼❪✱ ✶✾✽✾✳ ❆♥❞r❡❛s ❇❧❛ss ❛♥❞ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✳ ❬❇s❙❤✿✷✺✼❪✳ ❯❧tr❛✜❧t❡rs ✇✐t❤ s♠❛❧❧ ❣❡♥❡r❛t✐♥❣ s❡ts✳ ■sr❛❡❧ ❏✳ ▼❛t❤✳✱ ✻✺✿✷✺✾✕✷✼✶✱ ❬❇s❙❤✿✷✺✼❪✱ ✶✾✽✾✳ ❏ör❣ ❇r❡♥❞❧❡✳ ❉✐st✐♥❣✉✐s❤✐♥❣ ●r♦✉♣✇✐s❡ ❉❡♥s✐t② ◆✉♠❜❡rs✳ Pr❡♣r✐♥t✱ ✷✵✵✻✳ ▼❛rt✐♥ ●♦❧❞st❡r♥ ❛♥❞ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✳ ❘❛♠s❡② ✉❧tr❛✜❧t❡rs ❛♥❞ t❤❡ r❡❛♣✐♥❣ ♥✉♠❜❡r✖ ❈♦♥(r < u)✳ ❆♥♥✳ P✉r❡ ❆♣♣❧✳ ▲♦❣✐❝✱ ✹✾✱ ✶✾✾✵✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s

❇✐❜❧✐♦❣r❛♣❤②✱ ♣❛rt ✹

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r✳ ●r♦✉♣✇✐s❡ ❞❡♥s❡ ❢❛♠✐❧✐❡s✳ ❆r❝❤✳ ▼❛t❤✳ ▲♦❣✐❝✱ ✹✵✿✾✸ ✕✶✶✷✱ ✷✵✵✵✳ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r✳ ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u < d✱ t♦ ❛♣♣❡❛r ✐♥ ❏❙▲ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r✳ ❖♥ t❤❡ ❣r♦✉♣✇✐s❡ ❞❡♥s✐t② ♥✉♠❜❡r ❢♦r ✜❧t❡rs✳ ❆❝t❛ ❯♥✐✈✳ ❈❛r♦❧✐♥❛❡ ✲ ▼❛t❤✳ ❡t P❤②s✳✱ ✹✻✿✺✺✕✻✸✱ ✷✵✵✺✳ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r✱ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✱ ❛♥❞ ❇♦❛③ ❚s❛❜❛♥✳ ❈♦✈❡r✐♥❣ t❤❡ ❇❛✐r❡ ❙♣❛❝❡ ✇✐t❤ ▼❡❛❣❡r ❙❡ts✱ ❬▼❞❙❤❚s✿✽✹✼❪✳ ❆♥♥✳ P✉r❡ ❆♣♣❧✳ ▲♦❣✐❝ ✶✹✵✿✻✵✕✼✶✱ ✷✵✵✻✳

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u