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❆ ❘❛♠s❡②✲❚❤❡♦r❡t✐❝ ◆♦t✐♦♥ ♦❢ ❋♦r❝✐♥❣

❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❙❡t❚♦♣ ◆♦✈✐ ❙❛❞ ❏✉❧② ✸✱ ✷✵✶✽

❚❤❡ k✲✈❛❧✉❡❞ ❜❧♦❝❦s✱ Fink

❉❡✜♥✐t✐♦♥

▲❡t k ∈ ω \ {0} ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ✭✶✮ ❋♦r a: ω → k + 1 ✇❡ ❧❡t supp(a) = {n ∈ ω : a(n) = 0}✳ Fink = {a: ω → k + 1 : supp(a) ✐s ✜♥✐t❡ ∧ k ∈ range(a)}. ✭✷✮ ✳ ✭✸✮ ❋♦r ✱ ✇❡ ❧❡t ❞❡♥♦t❡ ✱ ✐✳❡✳✱ ✳ ❆ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ ✐s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥ ✐❢ ❢♦r ❛♥② ✱ ✳ ❚❤❡ s❡t ✐s t❤❡ s❡t ♦❢ ✲s❡q✉❡♥❝❡s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ ❜❧♦❝❦ s❡q✉❡♥❝❡s✳

❚❤❡ k✲✈❛❧✉❡❞ ❜❧♦❝❦s✱ Fink

❉❡✜♥✐t✐♦♥

▲❡t k ∈ ω \ {0} ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ✭✶✮ ❋♦r a: ω → k + 1 ✇❡ ❧❡t supp(a) = {n ∈ ω : a(n) = 0}✳ Fink = {a: ω → k + 1 : supp(a) ✐s ✜♥✐t❡ ∧ k ∈ range(a)}. ✭✷✮ Fin[1,k] = k

j=1 Finj✳

✭✸✮ ❋♦r ✱ ✇❡ ❧❡t ❞❡♥♦t❡ ✱ ✐✳❡✳✱ ✳ ❆ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ ✐s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥ ✐❢ ❢♦r ❛♥② ✱ ✳ ❚❤❡ s❡t ✐s t❤❡ s❡t ♦❢ ✲s❡q✉❡♥❝❡s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ ❜❧♦❝❦ s❡q✉❡♥❝❡s✳

❚❤❡ k✲✈❛❧✉❡❞ ❜❧♦❝❦s✱ Fink

❉❡✜♥✐t✐♦♥

▲❡t k ∈ ω \ {0} ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ✭✶✮ ❋♦r a: ω → k + 1 ✇❡ ❧❡t supp(a) = {n ∈ ω : a(n) = 0}✳ Fink = {a: ω → k + 1 : supp(a) ✐s ✜♥✐t❡ ∧ k ∈ range(a)}. ✭✷✮ Fin[1,k] = k

j=1 Finj✳

✭✸✮ ❋♦r a, b ∈ Fink✱ ✇❡ ❧❡t a < b ❞❡♥♦t❡ supp(a) < supp(b)✱ ✐✳❡✳✱ (∀m ∈ supp(a))(∀n ∈ supp(b))(m < n)✳ ❆ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ai : i < m ≤ ω ♦❢ ❡❧❡♠❡♥ts ♦❢ Fink ✐s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥ ✐❢ ❢♦r ❛♥② i < j < m✱ ai < aj✳ ❚❤❡ s❡t (Fink)ω ✐s t❤❡ s❡t ♦❢ ω✲s❡q✉❡♥❝❡s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ ❜❧♦❝❦ s❡q✉❡♥❝❡s✳

❚✇♦ ♦♣❡r❛t✐♦♥s ♦♥ Finj

❉❡✜♥✐t✐♦♥

✭✹✮ ❋♦r k ≥ 1✱ a, b ∈ Fink✱ ✇❡ ❞❡✜♥❡ t❤❡ ♣❛rt✐❛❧ s❡♠✐❣r♦✉♣ ♦♣❡r❛t✐♦♥ + ❛s ❢♦❧❧♦✇s✿ ■❢ supp(a) < supp(b)✱ t❤❡♥ a + b ∈ Fink ✐s ❞❡✜♥❡❞✳ ❲❡ ❧❡t (a + b)(n) = a(n) + b(n)✳ ❖t❤❡r✇✐s❡ a + b ✐s ✉♥❞❡✜♥❡❞✳ ❚❤✉s a+b = a ↾ supp(a)∪b ↾ supp(b)∪0 ↾ (ω\(supp(a)∪supp(b)))✳ ✭✺✮ ❋♦r ❛♥② ✇❡ ❞❡✜♥❡ ♦♥ t❤❡ ❚❡tr✐s ♦♣❡r❛t✐♦♥✿ ❜② ✳

❚✇♦ ♦♣❡r❛t✐♦♥s ♦♥ Finj

❉❡✜♥✐t✐♦♥

✭✹✮ ❋♦r k ≥ 1✱ a, b ∈ Fink✱ ✇❡ ❞❡✜♥❡ t❤❡ ♣❛rt✐❛❧ s❡♠✐❣r♦✉♣ ♦♣❡r❛t✐♦♥ + ❛s ❢♦❧❧♦✇s✿ ■❢ supp(a) < supp(b)✱ t❤❡♥ a + b ∈ Fink ✐s ❞❡✜♥❡❞✳ ❲❡ ❧❡t (a + b)(n) = a(n) + b(n)✳ ❖t❤❡r✇✐s❡ a + b ✐s ✉♥❞❡✜♥❡❞✳ ❚❤✉s a+b = a ↾ supp(a)∪b ↾ supp(b)∪0 ↾ (ω\(supp(a)∪supp(b)))✳ ✭✺✮ ❋♦r ❛♥② k ≥ 2 ✇❡ ❞❡✜♥❡ ♦♥ Fink t❤❡ ❚❡tr✐s ♦♣❡r❛t✐♦♥✿ T : Fink → Fink−1 ❜② T(a)(n) = max{a(n) − 1, 0}✳

• ❡♥❡r❛t❡❞ s❡♠✐❣r♦✉♣s

❉❡✜♥✐t✐♦♥

✭✻✮ ▲❡t B ⊆ Fink ❜❡ ♠✐♥✲✉♥❜♦✉♥❞❡❞✱ ✐✳❡✳✱ ❝♦♥t❛✐♥ ❢♦r ❛♥② n s♦♠❡ a ✇✐t❤ supp(a) > n✳ ❲❡ ❧❡t TFUk(B) ={T (j0)(bn0) + · · · + T (jℓ)(bnℓ) : ℓ ∈ ω \ {0}, bni ∈ B, bn0 < · · · < bnℓ, ji ∈ k, ∃r ≤ ℓjr = 0} ❜❡ t❤❡ ♣❛rt✐❛❧ s✉❜s❡♠✐❣r♦✉♣ ♦❢ Fink ❣❡♥❡r❛t❡❞ ❜② B✳ ❲❡ ❝❛❧❧ B ❛ TFUk✲s❡t ✐❢ B = TFUk(B)✳

❚❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r

❉❡✜♥✐t✐♦♥

✭✼✮ ❲❡ ❞❡✜♥❡ t❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r✿ ¯ a ⊑k ¯ b ✐❢ ¯ a ∈ (TFUk(¯ b))ω✳ ✭✽✮ ❲❡ ❞❡✜♥❡ t❤❡ ✲♦♣❡r❛t✐♦♥✿ ▲❡t ❛♥❞ ✳ ✇✐t❤ ✳

❚❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r

❉❡✜♥✐t✐♦♥

✭✼✮ ❲❡ ❞❡✜♥❡ t❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r✿ ¯ a ⊑k ¯ b ✐❢ ¯ a ∈ (TFUk(¯ b))ω✳ ✭✽✮ ❲❡ ❞❡✜♥❡ t❤❡ past ✲♦♣❡r❛t✐♦♥✿ ▲❡t ¯ a ∈ (Fink)ω ❛♥❞ s ∈ Fink✳ (¯ a past s) = ai : i ≥ i0 ✇✐t❤ i0 = min{i : supp(ai) > supp(s)}✳

❆ ♥❡❣❛t✐♦♥ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❢♦r ♥♦t ♥❡❝❡ss❛r✐❧② ❝❡♥tr❡❞ ❢❛♠✐❧✐❡s

❉❡✜♥✐t✐♦♥

✶✳ ❚✇♦ s✉❜s❡ts F1✱ F2 ♦❢ [ω]ω ❛r❡ ❝❛❧❧❡❞ ♥♥❝✱ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ ❢♦r ❛♥② Xi ∈ Fi✱ i = 1, 2 ❛♥❞ ❛♥② ✜♥✐t❡✲t♦✲♦♥❡ h: ω → ω t❤❡r❡ ✐s Yi ⊆ Xi✱ Yi ∈ Fi✱ i = 1, 2 s✉❝❤ t❤❛t h[Y1] ∩ h[Y2] = ∅✳ ✷✳ ▲❡t ❛♥❞ ❧❡t ❜❡ ❛ ✲♣♦✐♥t✳ ❲❡ s❛② ❛✈♦✐❞s ✐❢ ✐s ♥♥❝ t♦ ✳

❆ ♥❡❣❛t✐♦♥ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❢♦r ♥♦t ♥❡❝❡ss❛r✐❧② ❝❡♥tr❡❞ ❢❛♠✐❧✐❡s

❉❡✜♥✐t✐♦♥

✶✳ ❚✇♦ s✉❜s❡ts F1✱ F2 ♦❢ [ω]ω ❛r❡ ❝❛❧❧❡❞ ♥♥❝✱ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ ❢♦r ❛♥② Xi ∈ Fi✱ i = 1, 2 ❛♥❞ ❛♥② ✜♥✐t❡✲t♦✲♦♥❡ h: ω → ω t❤❡r❡ ✐s Yi ⊆ Xi✱ Yi ∈ Fi✱ i = 1, 2 s✉❝❤ t❤❛t h[Y1] ∩ h[Y2] = ∅✳ ✷✳ ▲❡t H ⊆ (Fink)ω ❛♥❞ ❧❡t E ❜❡ ❛ P✲♣♦✐♥t✳ ❲❡ s❛② H ❛✈♦✐❞s E ✐❢ {supp(¯ a) : ¯ a ∈ H} ✐s ♥♥❝ t♦ E✳

❆ s✉❜s♣❛❝❡ ♦❢ (Fink)ω✕❋✐①✐♥❣ PP ❛♥❞ ¯ R

❉❡✜♥✐t✐♦♥

❲❡ ✜① ♣❛r❛♠❡t❡rs ❛s ❢♦❧❧♦✇s✳ ▲❡t k ≥ 1✳ ❋✐① Pmin, Pmax ⊆ {1, . . . , k}✳ ▲❡t PP = {(i, x) : x ∈ {min, max}, i ∈ Px} ❛♥❞ ❧❡t ¯ R = {(ι, Rι) : ι ∈ PP} ❜❡ ❛ PP✲s❡q✉❡♥❝❡ ♦❢ ♣❛✐r✇✐s❡ ♥♥❝ ❘❛♠s❡② ✉❧tr❛✜❧t❡rs ✭♣❛✐r✇✐s❡ ♥♥❝ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧s✱ ✐✳❡✳✱ ❤❛♣♣② ❢❛♠✐❧✐❡s✱ ✇♦✉❧❞ s✉✣❝❡ ❢♦r t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt② ❛♥❞ ♣r♦♣❡r♥❡ss✮✳ ❲❡ ❛❧s♦ ♥❛♠❡ t❤❡ ❡♥❞ s❡❣♠❡♥ts ❢♦r 1 ≤ j ≤ k✿ ¯ R ↾ {j, . . . , k} = {(ι, Rι) : ι = (i, x) ∈ PP ∧ i ∈ {j, . . . , k}}.

❍❛♣♣② ❢❛♠✐❧✐❡s ✕ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧s

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ H ⊆ [ω]ω ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ ✐❢ ✶✳ ❛♥② ❝♦✜♥✐t❡ s✉❜s❡t ♦❢ ω ✐s ✐♥ H✱ ✷✳ ∀X ∈ H∀X1, X2(X1 ∪ X2 = X → X1 ∈ H ∨ X2 ∈ H)✳ ✸✳ ❋♦r ❛♥② An : n < ω s✉❝❤ t❤❛t ❢♦r ❛♥② n✱ An ∈ H ❛♥❞ An+1 ⊆ An t❤❡r❡ ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ A ∈ H✱ ✐✳❡✳✱ ∀n ∈ A(A \ (n + 1) ⊆ An). ❆ ❘❛♠s❡② ✉❧tr❛✜❧t❡r ✐s ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ t❤❛t ✐s ❛❧s♦ ❛ ✜❧t❡r✳

❍❛♣♣② ❢❛♠✐❧✐❡s ✕ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧s

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ H ⊆ [ω]ω ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ ✐❢ ✶✳ ❛♥② ❝♦✜♥✐t❡ s✉❜s❡t ♦❢ ω ✐s ✐♥ H✱ ✷✳ ∀X ∈ H∀X1, X2(X1 ∪ X2 = X → X1 ∈ H ∨ X2 ∈ H)✳ ✸✳ ❋♦r ❛♥② An : n < ω s✉❝❤ t❤❛t ❢♦r ❛♥② n✱ An ∈ H ❛♥❞ An+1 ⊆ An t❤❡r❡ ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ A ∈ H✱ ✐✳❡✳✱ ∀n ∈ A(A \ (n + 1) ⊆ An). ❆ ❘❛♠s❡② ✉❧tr❛✜❧t❡r ✐s ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ t❤❛t ✐s ❛❧s♦ ❛ ✜❧t❡r✳

❆ s✉❜s♣❛❝❡ ♦❢ (Fink)ω✿ ❚❤❡ s♣❛❝❡ (Fink)ω( ¯ R)

❉❡✜♥✐t✐♦♥

❲❡ ❧❡t (Fink)ω( ¯ R) ❞❡♥♦t❡ t❤❡ s❡t ♦❢ Fink✲❜❧♦❝❦s❡q✉❡♥❝❡s ¯ a ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

◮ (∀i ∈ Pmin){min(a−1 n [{i}]) : n ∈ ω} ∈ Ri,min✱ ◮ (∀i ∈ Pmax){max(a−1 n [{i}]) : n ∈ ω} ∈ Ri,max✱ ◮

(∀s ∈ TFUk(¯ a))

• min(s−1[{1}]) < min(s−1[{2}]) < · · · <

min(s−1[{k − 1}] < min(s−1[{k}]) < max(s−1[{k}]) < max(s−1[{k − 1}]) < · · · < max(s−1[{1}])

• .

■❢ (i, x) ∈ {1, . . . , k} × {min, max} \ PP✱ ✇❡ ❧❡❛✈❡ t❤❡ t❡r♠ x(s−1[{i}]) ♦✉t ♦❢ t❤❡ ♦r❞❡r r❡q✉✐r❡♠❡♥t✳

❲❡ ❞♦ ♥♦t ❧♦❝❛❧✐s❡ t♦ ❛ ❝❡♥tr❡❞ s❡t

▲❡♠♠❛

❚❤❡r❡ ❛r❡ ⊑∗

k✲✐♥❝♦♠♣❛t✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ (Fink)ω( ¯

R)✳ ■♥❞❡❡❞✱ t❤❡r❡ ❛r❡ ¯ a✱ ¯ b ∈ (Fink)ω( ¯ R) s✉❝❤ t❤❛t ❢♦r ❛♥② j = 0, . . . , k − 1 t❤❡ Fink−j✲❜❧♦❝❦✲s❡q✉❡♥❝❡s T (j)[¯ a] ❛♥❞ T (j)[¯ b] ❛r❡ ⊑∗

k−j✲✐♥❝♦♠♣❛t✐❜❧❡✳

❆ ❝♦♠♠♦♥ str❡♥❣t❤❡♥✐♥❣ ♦❢ ❛ t❤❡♦r❡♠ ❜② ●♦✇❡rs ❛♥❞ ❛ t❤❡♦r❡♠ ❜② ❇❧❛ss

❚❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ PP = {(1, min), (1, max)} ✇❛s ♣r♦✈❡❞ ❜② ❇❧❛ss ✐♥ ✶✾✽✼✱ t❤❡ ❝❛s❡ PP = ∅ ❛♥❞ ❛r❜✐tr❛r② ✜♥✐t❡ k ❜② ●♦✇❡rs ✐♥ ✶✾✾✷✳

❚❤❡♦r❡♠

▲❡t k✱ PP✱ ¯ R ❜❡ ❛s ❛❜♦✈❡✳ ▲❡t ¯ a ∈ (Fink)ω( ¯ R) ❛♥❞ ❧❡t c ❜❡ ❛ ❝♦❧♦✉r✐♥❣ ♦❢ TFUk(¯ a) ✐♥t♦ ✜♥✐t❡❧② ♠❛♥② ❝♦❧♦✉rs✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ¯ b ⊑k ¯ a✱ ¯ b ∈ (Fink)ω( ¯ R)✱ s✉❝❤ t❤❛t TFUk(¯ b) ✐s c✲♠♦♥♦❝❤r♦♠❛t✐❝✳

❉✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞s

▲❡♠♠❛

❧❡t k✱ PP✱ ¯ R ❜❡ ❛s ❛❜♦✈❡✳ ❆♥② ⊑k✲❞❡s❝❡♥❞✐♥❣ s❡q✉❡♥❝❡ ¯ cn : n ∈ ω ✐♥ (Fink)ω( ¯ R) ❤❛s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ ¯ b ∈ (Fink)ω( ¯ R) (∀n ∈ ω)((¯ b past bn) ⊑k ¯ cmax(supp(bn))+1). s✉❝❤ t❤❛t ❡❛❝❤ bn+1 ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ {cℓn+1,m : m ∈ ω} ❢♦r s♦♠❡ ℓn+1 > max(supp(bn)) ❛♥❞ b0 ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ {cℓ0,m : m ∈ ω} ❢♦r s♦♠❡ ℓ0✳

❆ k✲st❛❝❦ ♦❢ ❝♦♠♣❛❝t s♣❛❝❡s

γ(Finj( ¯ R ↾ {k + j − 1, . . . , k})) ✐s t❤❡ s❡t ♦❢ ✉❧tr❛✜❧t❡rs U ♦✈❡r Finj s✉❝❤ t❤❛t ❢♦r ❛♥② ¯ a ∈ U✱ ℓ ∈ {1, . . . , j}✱ {min(a−1

n [{ℓ}]) : n ∈ ω} ∈ Rℓ+k−j,min

❛♥❞ ❛♥❛❧♦❣♦✉s❧② ❢♦r max✳

• ❛❧✈✐♥✕●❧❛③❡r t❡❝❤♥✐q✉❡

❉❡✜♥✐t✐♦♥

❋♦r ❛♥② k ≥ 1✱ ❛ r❡s❡r✈♦✐r ♦❢ ✐♥❞✐❝❡s PP ♦❢ t❤❡ str✐❝t ❢♦r♠ ✐s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ t②♣❡s✿ PP = {(i, min), (i, max) : 1 ≤ i ≤ k}✱ PP = {(i, min) : 1 ≤ i ≤ k}✱ PP = {(i, min) : 1 ≤ i ≤ k}✳

❉❡✜♥✐t✐♦♥ ❛♥❞ ▲❡♠♠❛

❍❡r❡ ✇❡ ❧❡t PP ❜❡ ♦❢ t❤❡ str✐❝t ❢♦r♠✳ ❲❡ ❞❡✜♥❡ ˙ + ♦♥ (k

j=1 γ(Finj( ¯

R ↾ {k − j + 1, . . . , k})))2 ❛s ❢♦❧❧♦✇s✳ ˙ +: γ(Fini( ¯ R ↾ {k − i + 1, . . . , k})) × γ(Finj( ¯ R ↾ {k − j + 1, . . . , k})) → γ(Finmax{i,j}( ¯ R ↾ {k − max(i, j) + 1, . . . , k})) ✐s ❞❡✜♥❡❞ ❛s U ˙ +V =

• X ⊆ Finmax{i,j}( ¯

R ↾ {k − max(i, j) + 1, . . . , k}) :

• s : {t : s + t ∈ X} ∈ V
• ∈ U
• .

❆ k✲s❡q✉❡♥❝❡ ♦❢ ✈❡r② ❣♦♦❞ ✐❞❡♠♣♦t❡♥t ✉❧tr❛✜❧t❡rs

▲❡♠♠❛

❙t✐❧❧ PP ♦❢ t❤❡ str✐❝t ❢♦r♠✳ ✭▲❡♠♠❛ ✷✳✷✹✱ ❚♦❞♦r❝❡✈✐❝✱ ❘❛♠s❡② ❙♣❛❝❡s✮ ▲❡t k✱ PP✱ ¯ R ❜❡ ❛s ❛❜♦✈❡✱ ✇✐t❤ ❢✉❧❧ PP✳ ❋♦r ❛♥② k ≥ j ≥ 1✱ ❛♥❞ ¯ a ∈ (Fink)ω( ¯ R) t❤❡r❡ ✐s ❛♥ ✐❞❡♠♣♦t❡♥t Uj ∈ γ(Finj( ¯ R ↾ {k + j − 1, . . . , k})) s✉❝❤ t❤❛t ❢♦r ❛❧❧ 1 ≤ i ≤ j ≤ k ✭✶✮ Uj ˙ +Ui = Uj✱ ✭✷✮ ˙ T (j−i)(Uj) = Ui✳ ✭✸✮ T (i−1)(¯ a) ∈ Uk−i+1✳

❙t❡♣♣✐♥❣ ✉♣ t♦ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥s

❙✐♥❝❡ t❤❡ s♣❛❝❡ (Fink)ω( ¯ R) ✐s st❛❜❧❡✱ ✇❡ ❝❛♥ st❡♣ ✉♣ t❤❡ ▼✐❧❧✐❦❡♥✕❚❛②❧♦r st②❧❡ t♦ ❤✐❣❤❡r ✜♥✐t❡ ❛r✐t✐❡s✿

❚❤❡♦r❡♠

▲❡t n ∈ ω \ {0} ❛♥❞ ¯ a ∈ (Fink)ω( ¯ R) ❛♥❞ ❧❡t c ❜❡ ❛ ❝♦❧♦✉r✐♥❣ ♦❢ (TFUk(¯ a))n

< ✐♥t♦ ✜♥✐t❡❧② ♠❛♥② ❝♦❧♦✉rs✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ¯

b ⊑k ¯ a✱ ¯ b ∈ (Fink)ω( ¯ R) s✉❝❤ t❤❛t (TFUk(¯ b))n

< ✐s c✲♠♦♥♦❝❤r♦♠❛t✐❝✳

❆ ✉s❡❢✉❧ ♥♦t✐♦♥ ♦❢ ❢♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

❲❡ ❧❡t k✱ PP✱ ¯ R ❜❡ ❛s ❛❜♦✈❡✱ ♥♦t ♥❡❝❡ss❛r✐❧② str✐❝t✳ ■♥ t❤❡

• ♦✇❡rs✕▼❛t❡t ❢♦r❝✐♥❣ ✇✐t❤ ¯

R✱ Mk( ¯ R)✱ t❤❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♣❛✐rs (s, ¯ c) s✉❝❤ t❤❛t s ∈ Fink ❛♥❞ ¯ c ∈ (Fink)ω( ¯ R) ❛♥❞ supp(s) < supp(c0)✳ ❚❤❡ ❢♦r❝✐♥❣ ♦r❞❡r ✐s✿ (t,¯ b) ≤ (s, ¯ a) ✐❢ t = s + s′ ❛♥❞ s′ ∈ TFUk(¯ a) ❛♥❞ ¯ b ⊑k (¯ a past s′)

▲❡♠♠❛

❤❛s t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✱ ✐✳❡✳✱ ❢♦r ❛♥② ✱ ✳

❆ ✉s❡❢✉❧ ♥♦t✐♦♥ ♦❢ ❢♦r❝✐♥❣

❉❡✜♥✐t✐♦♥

❲❡ ❧❡t k✱ PP✱ ¯ R ❜❡ ❛s ❛❜♦✈❡✱ ♥♦t ♥❡❝❡ss❛r✐❧② str✐❝t✳ ■♥ t❤❡

• ♦✇❡rs✕▼❛t❡t ❢♦r❝✐♥❣ ✇✐t❤ ¯

R✱ Mk( ¯ R)✱ t❤❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♣❛✐rs (s, ¯ c) s✉❝❤ t❤❛t s ∈ Fink ❛♥❞ ¯ c ∈ (Fink)ω( ¯ R) ❛♥❞ supp(s) < supp(c0)✳ ❚❤❡ ❢♦r❝✐♥❣ ♦r❞❡r ✐s✿ (t,¯ b) ≤ (s, ¯ a) ✐❢ t = s + s′ ❛♥❞ s′ ∈ TFUk(¯ a) ❛♥❞ ¯ b ⊑k (¯ a past s′)

▲❡♠♠❛

Mk( ¯ R) ❤❛s t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✱ ✐✳❡✳✱ ❢♦r ❛♥② ϕ ∈ L(∈)✱ (s, ¯ a) ∈ Mk( ¯ R) ∃(s,¯ b) ≤ (s, ¯ a) ((s,¯ b) ϕ ∨ (s,¯ b) ¬ϕ)✳

• ♦♦❞ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡s❡r✈♦✐rs ♦❢ ♣✉r❡ ❝♦♥❞✐t✐♦♥s

❉❡✜♥✐t✐♦♥

❆ s❡t H ⊆ (Fink)ω ✐s ❝❛❧❧❡❞ ❛ ●♦✇❡rs✕▼❛t❡t✲❛❞❡q✉❛t❡ ❢❛♠✐❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿ ✶✳ H ✐s ❝❧♦s❡❞ ⊑∗

k✲✉♣✇❛r❞s✳

✷✳ H ✐s st❛❜❧❡✱ ✐✳❡✳✱ ❛♥② ⊑k✲❞❡s❝❡♥❞✐♥❣ ω✲s❡q✉❡♥❝❡ ♦❢ ♠❡♠❜❡rs ♦❢ H ❤❛s ❛ ⊑∗ ❧♦✇❡r ❜♦✉♥❞ ✐♥ H✳ ✸✳ H ❤❛s t❤❡ ●♦✇❡rs ♣r♦♣❡rt②✿ ■❢ ¯ a ∈ H ❛♥❞ TFUk(¯ a) ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ✜♥✐t❡❧② ♠❛♥② ♣✐❡❝❡s t❤❡♥ t❤❡r❡ ✐s s♦♠❡ ¯ b ⊑k ¯ a✱ ¯ b ∈ H s✉❝❤ t❤❛t TFUk(¯ b) ✐s ❛ s✉❜s❡t ♦❢ ❛ s✐♥❣❧❡ ♣✐❡❝❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥✳

❊①❛♠♣❧❡s ♦❢ ●♦✇❡rs✕▼❛t❡t✲❛❞❡q✉❛t❡ ❢❛♠✐❧✐❡s

◮ H = (Fink)ω( ¯

R)✱ ✇❡ ✇r✐t❡ M( ¯ R) ❢♦r M(H)✳

◮ M(U) ❢♦r ❛ ●♦✇❡rs✲▼✐❧❧✐❦❡♥✲❚❛②❧♦r ✉❧tr❛✜❧t❡r✳ ◮ ■♥st❡❛❞ ♦❢ ✐♠♣♦s✐♥❣ t❤❛t mini[¯

a]✱ maxi[¯ a] ❝♦♠❡ ❢r♦♠ ❤❛♣♣② ❢❛♠✐❧✐❡s ✇❤❡♥ ¯ a ∈ H ✇❡ ❝♦✉❧❞ tr② t♦ ✉s❡ seti(¯ a) ❢♦r i ∈ {1, . . . , k}✳

❚❤❡ ❢❛t❡ ♦❢ t❤❡ Ri,x✱ (i, x) ∈ PP✱ ✐♥ VM( ¯

R)

❉❡✜♥✐t✐♦♥

❚❤❡ i✲✜❜r❡ ♦❢ t❤❡ ❣❡♥❡r✐❝ r❡❛❧ µ = {s ↾ supp(s) : ∃¯ a(s, ¯ a) ∈ G} ✐s µi =

• {s−1[{i}] : ∃¯

a(s, ¯ a) ∈ G}, supp(µ) ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ µi✳ ❉❡♥s✐t② ❛r❣✉♠❡♥t✿ µi ✐s ♥♦t ♠❡❛s✉r❡❞ ❜② Ri,min✱ Ri,max✳

❉❡✜♥✐t✐♦♥

▲❡t X ∈ [ω]ω✳ ❲❡ ❧❡t fX(n) = |X ∩ n|✳

❚❤❡ ❢❛t❡ ♦❢ ♦t❤❡r ✉❧tr❛✜❧t❡rs ✐♥ VM( ¯

R)

▲❡♠♠❛

▲❡t h: ω → ω ❜❡ ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥✳ ▲❡t E ❛♥❞ W ❜❡ ✉❧tr❛✜❧t❡rs ♦✈❡r ω s✉❝❤ t❤❛t W✱ E ≥RB Rι ❢♦r ι ∈ PP✳ ❚❤❡♥ ❢♦r ❛♥② (s, ¯ a) ∈ Mk( ¯ R)✱ E ∈ E t❤❡r❡ ❛r❡ ¯ b ⊑k ¯ a✱ ¯ b ∈ (Fink)ω( ¯ R)✱ ❛♥❞ E′ ∈ E✱ E′ ⊆ E ❛♥❞ W ∈ W s✉❝❤ t❤❛t ✭✶✮ h[{supp(bn) : n ∈ ω}] ∩ h[E′] = ∅✳ ✭✷✮ h[{[min(supp(bn)), max(supp(bn))] : n ∈ ω}] ∩(h[E′] ∪ h[W]) = ∅✱ ❛♥❞ (s,¯ b) Mk( ¯

R) fsupp(µ)[W] = fsupp(µ)[E′]✳

❋r✉✐t ♦❢ ❝♦♥❝❧✉s✐♦♥ ✶ ♦❢ t❤❡ ❧❡♠♠❛

❚❤❡♦r❡♠

✭❆❞❛♣t✐♦♥ ♦❢ ❛ t❤❡♦r❡♠ ♦❢ ❊✐s✇♦rt❤✮ ▲❡t k ≥ 1 ❛♥❞ ¯ R ❜❡ ❛s ❛❜♦✈❡ ❛♥❞ ❛ss✉♠❡ t❤❛t E ✐s ❛ P✲♣♦✐♥t ✇✐t❤ E ≥RB R(i,min), R(j,max) ❢♦r ❛♥② i ∈ Pmin ❛♥❞ j ∈ Pmax✳ ❚❤❡♥ E ❝♦♥t✐♥✉❡s t♦ ❣❡♥❡r❛t❡ ❛♥ ✉❧tr❛✜❧t❡r ❛❢t❡r ✇❡ ❢♦r❝❡ ✇✐t❤ Mk( ¯ R)✳

❋r✉✐t ♦❢ ❝♦♥❝❧✉s✐♦♥ ✷ ♦❢ t❤❡ ❧❡♠♠❛

❚❤❡♦r❡♠

▲❡t k ≥ 1 ❛♥❞ ¯ R ❜❡ ❛s ❛❜♦✈❡ ❛♥❞ ❛ss✉♠❡ E, W ≥RB R(i,min), R(j,max) ❢♦r ❛♥② i ∈ Pmin ❛♥❞ j ∈ Pmax ❛♥❞ ❧❡t E ❜❡ ❛ P✲♣♦✐♥t ❛♥❞ W ❜❡ ❛♥ ✉❧tr❛✜❧t❡r ♦✈❡r ω✳ ❚❤❡♥ Mk( ¯ R) fsupp(µ)(E) = fsupp(µ)(W).

❙t❛rt ♦❢ ❛♥ ✭❝s✮ ✐t❡r❛t✐♦♥

◆♦✇ ✇❡ ❛r❡ ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ s❡❝♦♥❞ ✐t❡r❛♥❞✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦❧❧♦✇s ❢r♦♠ ❛♥ ❡❛s② ❞❡♥s✐t② ❛r❣✉♠❡♥t✳

▲❡♠♠❛

▲❡t ι = (i, x) ∈ PP✳ Mk( ¯ R) Rι ∪ {µi} ✐s ❛ ✜❧t❡r s✉❜❜❛s❡✳

❋✐♥❞✐♥❣ ❛ s❡❝♦♥❞ ✐t❡r❛♥❞

❚❤❡♦r❡♠

▲❡t k✱ PP✱ ¯ R ❜❡ ❛s ✐♥ t❤❡ ♥♦♥✲str✐❝t ❢♦r♠✱ ι ∈ PP✳ Mk( ¯ R) (filter((Rι ∪ {µi}))+ ✐s ❛ ❤❛♣♣② ❢❛♠✐❧② t❤❛t ❛✈♦✐❞s E ❛♥❞ ❢♦r ι = ι′ t❤❡ ❢❛♠✐❧② (filter((Rι ∪ {µi}))+ ✐s ♥♥❝ t♦ t❤❡ ❢❛♠✐❧② (filter((Rι′ ∪ {µi′}))+✳ ❛♥❞ ❤❡♥❝❡ Mk( ¯ R) (∃Rext

ι

⊇ (Rι ∪ {µi})

• Rext

ι

✐s ❛ ❘❛♠s❡② ✉❧tr❛✜❧t❡r t❤❛t ✐s ♥♥❝ t♦ E ❛♥❞ ❢♦r ι = ι′✱ Rext

ι

♥♥❝ Rext

ι′

• .

❘❛♠s❡② s♣❛❝❡s ♦❢ ♥❛♠❡s

▲❡♠♠❛

✭❊①✐st❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞s✮ ▲❡t U ❜❡ ❛♥ ▼✐❧❧✐❦❡♥✕❚❛②❧♦r ✉❧tr❛✜❧t❡r✱ E ❜❡ ❛ P✲♣♦✐♥t✱ Φ(U) ≤RB E✳ ▲❡t Q = M(U) ❛♥❞ ❧❡t µ ❜❡ t❤❡ ♥❛♠❡ ❢♦r t❤❡ ❣❡♥❡r✐❝ r❡❛❧✳ ▲❡t ¯ X ˜ = Xn ˜ : n ∈ ω ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ Q✲♥❛♠❡s ❢♦r ❡❧❡♠❡♥ts ♦❢ (Fin)ω s✉❝❤ t❤❛t Q (∀n ∈ ω)(Xn ˜ ∈ (U ↾ µ)+ ∧ Xn+1 ˜ ⊑ Xn ˜ ).

❈♦♥t✐♥✉❡❞

▲❡♠♠❛ ❝♦♥t✐♥✉❡❞

❚❤❡♥ D ˜ =

• t,(s, ¯

a) : (s, ¯ a) ∈ Q ∧ (∃k ∈ ω)(∃t0 < t1 < · · · < tk−1 ∈ [Fin]k

<)

• tk−1 < tk = t ∧ (s, ¯

a) ✏t0 = min

Fin (X0

˜ ↾ µ)∧

• i<k

ti+1 = min

Fin ((Xmax(ti)+1

˜ ↾ µ) past ti)✑

• ❢✉❧✜❧s

Q D ˜ ∈ (U ↾ µ)+ ∧ D ˜ ⊑ X0 ˜ ∧ (∀t ∈ D ˜ )((D ˜ past t) ⊑ Xmax(t)+1 ˜ ).

❆ ❩❋❈ r❡s✉❧t

Pr♦♣♦s✐t✐♦♥

▲❡t E ❜❡ ❛ ✜❧t❡r ♦✈❡r ω✱ ❛♥❞ ❧❡t V ❛♥❞ W ❜❡ t✇♦ ✜❧t❡rs ♦✈❡r ω t❤❛t ❛r❡ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ E✳ ■❢ V ✐s ♥❡❛r❧② ❝♦❤❡r❡♥t t♦ W✱ t❤❡♥ t❤❡r❡ ✐s E ∈ E s✉❝❤ t❤❛t fE(V) ∪ fE(W) ✐s ❛ ✜❧t❡r s✉❜❜❛s❡✳

❈❛rr②✐♥❣ ♦♥ ✐♥ ❝s ❧✐♠✐t st❡♣s

❚❤❡♦r❡♠

❙✉♣♣♦s❡ t❤❛t Pβ, ¯ Rβ ❛r❡ ❛s ❛❜♦✈❡ Pα ✐s t❤❡ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ❧✐♠✐t ♦❢ Pβ, Mk( ¯ Rβ) : β < α✳ ■♥ VPα✱ ❢♦r ❛♥② ι ∈ PP✱ t❤❡ s❡t ♦❢ ♣♦s✐t✐✈❡ s❡ts

γ<α

(Rγ,ι ∪ {µγ,i}) + ❢♦r♠s ❛ ❤❛♣♣② ❢❛♠✐❧② t❤❛t ❛✈♦✐❞s E ❛♥❞ t❤❡ ❤❛♣♣② ❢❛♠✐❧✐❡s ❛r❡ ♣❛✐r✇✐s❡ ♥♥❝✳

◆❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✐♥ ❛♥ ✐t❡r❛t✐♦♥ ♦❢ ❧❡♥❣t❤ ω2

❚❤❡♦r❡♠

▲❡t E ❜❡ ❛ P✲♣♦✐♥t ❛♥❞ ❛ss✉♠❡ ❈❍ ❛♥❞ ❧❡t k ≥ 1 ❛♥❞ ❧❡t PP ⊆ {(i, x) : x = min, max, i = 1, . . . , k}✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ ❝♦✉♥t❛❜❧❡ s✉♣♣♦rt ✐t❡r❛t✐♦♥ ✐t❡r❛t✐♦♥ ♦❢ ♣r♦♣❡r ✐t❡r❛♥❞s P = Pα, Mk( ¯ Rβ) : β < ω2, α ≤ ω2 t❤❛t ✐♥ t❤❡ ❡①t❡♥s✐♦♥ t❤❡r❡ ❡①❛❝t❧② |PP| + 1 ♥❡❛r✲❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ♦❢ ✉❧tr❛✜❧t❡rs✳ ◆❛♠❡❧②✱ ♦♥❡ ❝❧❛ss ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ P✲♣♦✐♥t ♦❢ ❝❤❛r❛❝t❡r ω1 ❛♥❞ |PP| ❝❧❛ss❡s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❘❛♠s❡② ✉❧tr❛✜❧t❡rs Ri,x =

• {Ri,x,α : α < ω2},

(i, x) ∈ PP✳

❆ ❢❛❝t♦r✐s❛t✐♦♥

Pr♦♣♦s✐t✐♦♥

❲❡ ❧❡t Qpure = (Finω

k ( ¯

R), ⊑∗

k) ❛♥❞ ✇❡ ❧❡t

U ˜ = {¯ a, ˇ ¯ a : ¯ a ∈ Qpure}✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✭✶✮ Qpure ✐s ω✲❝❧♦s❡❞✳ ✭✷✮ Mk( ¯ R) ✐s ❞❡♥s❡❧② ❡♠❜❡❞❞❡❞ ✐♥t♦ Qpure ∗ Mk(U ˜ )✳ ✭✸✮ Qpure ❢♦r❝❡s t❤❛t U ˜ ✐s ❛ ●♦✇❡rs✕▼✐❧❧✐❦❡♥✕❚❛②❧♦r ✉❧tr❛✜❧t❡r ✇✐t❤ ˆ mini(U) = Ri,min ❛♥❞ ˆ maxj(U) = Rj,max✳ ✭✹✮ Qpure ❢♦r❝❡s t❤❛t Φ(U ˜ ) ✐s ♥♥❝ t♦ ❛♥② ✜❧t❡r ❢r♦♠ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ t❤❛t ✐s ♥♥❝ Rι✱ ι ∈ PP✳

Pr♦♦❢ ♦❢ ❛ ❝♦♥❥❡❝t✉r❡ ♦❢ ❇❧❛ss

■♥ ✶✾✽✼ ❇❧❛ss ❝♦♥❥❡❝t✉r❡❞ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ♦❢ t✇♦ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ❘❛♠s❡② ✉❧tr❛✜❧t❡rs ❞♦❡s ♥♦t ✐♠♣❧② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ▼✐❧❧✐❦❡♥✕❚❛②❧♦r ✉❧tr❛✜❧t❡r✳

❚❤❡♦r❡♠

❋♦r ❛♥② k✱ PP✱ ¯ R ✐♥ t❤❡ ❢♦r❝✐♥❣ ❡①t❡♥s✐♦♥s ❢r♦♠ t❤❡ ♠❛✐♥ t❤❡♦r❡♠✱ ✐s t❤❡r❡ ✐s ♥♦ ●♦✇❡rs✕▼✐❧❧✐❦❡♥✕❚❛②❧♦r ✉❧tr❛✜❧t❡r ♦✈❡r Fink′ ❢♦r ❛♥② k′ ≥ 1✳ ❘❡❛s♦♥✿ ■❢ V ✐s ❛♥ ▼✐❧❧✐❦❡♥✕❚❛②❧♦r ✉❧tr❛✜❧t❡r✱ t❤❡♥ t❤✐s ❤♦❧❞s ❢♦r Pα✲♣❛rt ✐♥ VPα ❢♦r ❝❧✉❜ ♠❛♥② α < ω2✳ ❯♥❞❡r ❈❍✱ t❤❡ ❝♦r❡ Φ(V) ∩ VPα ❝♦♥t❛✐♥s ❛ tr❡❡ ♦❢ 2ω1 ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s✳