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❙♣❤❡r✐❝❛❧ ❉❡s✐❣♥s ❛♥❞ ◆✉♠❡r✐❝❛❧ ❆♥❛❧②s✐s

❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q)

P✳ ❇♦②✈❛❧❡♥❦♦✈ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ❇✉❧❣❛r✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❏♦✐♥t ✇♦r❦ ✇✐t❤✿ P✳ ❉r❛❣♥❡✈✱ ✭❉❡♣t✳ ▼❛t❤✳ ❙❝✐❡♥❝❡s✱ ■P❋❲✱ ❋♦rt ❲❛②♥❡✱ ■◆✱ ❯❙❆✮ ❉✳ ❍❛r❞✐♥✱ ❊✳ ❙❛✛ ✭❉❡♣t✳ ▼❛t❤✳✱ ❱❛♥❞❡r❜✐❧t ❯♥✐✈❡rs✐t②✱ ◆❛s❤✈✐❧❧❡✱ ❚◆✱ ❯❙❆✮ ▼✳ ❙t♦②❛♥♦✈❛ ✭❋❛❝✳ ▼❛t❤✳ ■♥❢♦r♠✳✱ ❙♦✜❛ ❯♥✐✈❡rs✐t②✱ ❇✉❧❣❛r✐❛✮ ❙❤❛♥❣❤❛✐✱ ❈❤✐♥❛✱ ❆♣r✐❧ ✷✶✲✷✹✱ ✷✵✶✺

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶ ✴ ✸✽

❈♦♥t❡♥ts

❉❡✜♥✐t✐♦♥s ❛♥❞ ♥♦t❛t✐♦♥s ✭s♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s✱ ❡♥❡r❣②✱ ♣r♦❜❧❡♠s✱ s♦♠❡ r❡❢❡r❡♥❝❡s✮ Pr❡❧✐♠✐♥❛r✐❡s ✭❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞✱ ▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞✱ ✉s❡❢✉❧ q✉❛❞r❛t✉r❡✮ ▲P ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s✴❞❡s✐❣♥s ✭❢♦❧❦❧♦r❡✮ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❜♦✉♥❞ ✭❯▲❇✮ ❚✇♦ ✇❛②s ♦❢ ✐♠♣r♦✈✐♥❣ ❯▲❇ ❢♦r ❞❡s✐❣♥s ✕ s❤r✐♥❦✐♥❣ t❤❡ ✐♥t❡r✈❛❧ ✭❜♦✉♥❞s ♦♥ ✐♥♥❡r ♣r♦❞✉❝ts✱ ▲P ✐♥ s❤♦rt❡r ✐♥t❡r✈❛❧✮✱ ❛♥❞ ❤✐❣❤❡r ❞❡❣r❡❡s ✭t❡st✲❢✉♥❝t✐♦♥s✱ ♦♣t✐♠❛❧ ❝♦❞❡s✴❞❡s✐❣♥s✮ ❯♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❞❡s✐❣♥s ❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ✭❯▲❇ ❛♥❞ ▲P ✐♥ s❤♦rt❡r ✐♥t❡r✈❛❧✮ P❛r❛❧❧❡❧s ❢♦r ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ✐♥ ❍❛♠♠✐♥❣ s♣❛❝❡s

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷ ✴ ✸✽

❙♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s

▲❡t Sn−✶ ❞❡♥♦t❡ t❤❡ ✉♥✐t s♣❤❡r❡ ✐♥ Rn✳ ❲❡ r❡❢❡r t♦ ❛ ✜♥✐t❡ s❡t C ⊂ Sn−✶ ❛s ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡✳ ❆ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥ ✐s ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡ s✉❝❤ t❤❛t ✶ µ(Sn−✶)

• Sn−✶ f (x)dµ(x) = ✶

|C|

• x∈C

f (x) ✭µ(x) ✐s t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✮ ❤♦❧❞s ❢♦r ❛❧❧ ♣♦❧②♥♦♠✐❛❧s f (x) = f (x✶, x✷, . . . , xn) ♦❢ t♦t❛❧ ❞❡❣r❡❡ ❛t ♠♦st τ✳ ❚❤❡ ♠❛①✐♠❛❧ ♥✉♠❜❡r τ = τ(C) s✉❝❤ t❤❛t C ✐s ❛ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥ ✐s ❝❛❧❧❡❞ t❤❡ str❡♥❣t❤ ♦❢ C✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸ ✴ ✸✽

❊♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s✴❞❡s✐❣♥s

▲❡t h(t) : [−✶, ✶) → (✵, +∞) ❜❡ ❣✐✈❡♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ h✲❡♥❡r❣② ✭♦r ♣♦t❡♥t✐❛❧ ❡♥❡r❣②✮ ♦❢ C ✐s ❞❡✜♥❡❞ E(n, C; h) :=

• x,y∈C,x=y

h(x, y), ✇❤❡r❡ x, y ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢ x ❛♥❞ y✳ ❋♦r ♦✉r ♠❛✐♥ r❡s✉❧ts ✇❡ r❡q✉✐r❡ h t♦ ❜❡ ✭str✐❝t❧②✮ ❛❜s♦❧✉t❡❧② ♠♦♥♦t♦♥❡ ♦♥ ❬✲✶✱✶✮❀ ✐✳❡✳✱ t❤❡ k✲t❤ ❞❡r✐✈❛t✐✈❡ ♦❢ h s❛t✐s✜❡s h(k)(t) ≥ ✵ ✭h(k)(t) > ✵✮ ❢♦r ❛❧❧ k ≥ ✵ ❛♥❞ t ∈ [−✶, ✶)✳ ❆ ❝♦♠♠♦♥❧② ❛r✐s✐♥❣ ♣r♦❜❧❡♠ ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ♣r♦✈✐❞❡❞ t❤❡ ❝❛r❞✐♥❛❧✐t② |C| ♦❢ C ✐s ✜①❡❞❀ t❤❛t ✐s✱ t♦ ❞❡t❡r♠✐♥❡ E(n, N; h) := ✐♥❢{E(n, C; h) : |C| = N} t❤❡ ♠✐♥✐♠✉♠ ♣♦ss✐❜❧❡ h✲❡♥❡r❣② ♦❢ ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡ ♦❢ ❝❛r❞✐♥❛❧✐t② N✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✹ ✴ ✸✽

❊♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥s

▲❡t C ⊂ Sn−✶ ❜❡ ❛ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥ ❛♥❞ E(n, C; h) ❜❡ t❤❡ h✲❡♥❡r❣② ♦❢ C✳ ❉❡♥♦t❡ ❜② L(n, N, τ; h) = ✐♥❢{E(n, C; h) : |C| = N, C ⊂ Sn−✶, C ✐s τ✲❞❡s✐❣♥} t❤❡ ♠✐♥✐♠✉♠ ♣♦ss✐❜❧❡ h✲❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥s ♦♥ Sn−✶ ♦❢ N ♣♦✐♥ts✱ ❙✐♠✐❧❛r❧②✱ U(n, N, τ; h) = s✉♣{E(n, C; h) : |C| = N, C ⊂ Sn−✶, C ✐s τ✲❞❡s✐❣♥} t❤❡ ♠❛①✐♠✉♠ ♣♦ss✐❜❧❡ h✲❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥s ♦♥ Sn−✶ ♦❢ N ♣♦✐♥ts✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✺ ✴ ✸✽

❙♦♠❡ r❡❢❡r❡♥❝❡s

P✳ ❉❡❧s❛rt❡✱ ❆♥ ❆❧❣❡❜r❛✐❝ ❆♣♣r♦❛❝❤ t♦ t❤❡ ❆ss♦❝✐❛t✐♦♥ ❙❝❤❡♠❡s ✐♥ ❈♦❞✐♥❣ ❚❤❡♦r②✱ P❤✐❧✐♣s ❘❡s✳ ❘❡♣✳ ❙✉♣♣❧✳ ✶✵✱ ✭✶✾✼✸✮✳ P✳ ❉❡❧s❛rt❡✱ ❏✳✲▼✳ ●♦❡t❤❛❧s✱ ❏✳ ❏✳ ❙❡✐❞❡❧✱ ❙♣❤❡r✐❝❛❧ ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s✱

• ❡♦♠✳ ❉❡❞✐❝❛t❛ ✻✱ ✶✾✼✼✱ ✸✻✸✲✸✽✽✳
• ✳ ❆✳ ❑❛❜❛t✐❛♥s❦②✱ ❱✳ ■✳ ▲❡✈❡♥s❤t❡✐♥✱ ❇♦✉♥❞s ❢♦r ♣❛❝❦✐♥❣s ♦♥ ❛ s♣❤❡r❡

❛♥❞ ✐♥ s♣❛❝❡✱ Pr♦❜❧❡♠s ♦❢ ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ ✶✹✱ ✶✕✶✼✱ ✭✶✾✼✽✮✳ ❱✳ ■✳ ▲❡✈❡♥s❤t❡✐♥✱ ❯♥✐✈❡rs❛❧ ❜♦✉♥❞s ❢♦r ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s✱ ❍❛♥❞❜♦♦❦ ♦❢ ❈♦❞✐♥❣ ❚❤❡♦r②✱ ❱✳ ❙✳ P❧❡ss ❛♥❞ ❲✳ ❈✳ ❍✉✛♠❛♥✱ ❊❞s✳✱ ❊❧s❡✈✐❡r✱ ❆♠st❡r❞❛♠✱ ✶✾✾✽✱ ❈❤✳ ✻✱ ✹✾✾✕✻✹✽✳ ❱✳ ❆✳ ❨✉❞✐♥✱ ▼✐♥✐♠❛❧ ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ♦❢ ❛ ♣♦✐♥t s②st❡♠ ♦❢ ❝❤❛r❣❡s✱ ❉✐s❝r✳ ▼❛t❤✳ ❆♣♣❧✳ ✸✱ ✼✺✲✽✶✱ ✶✾✾✸✳ ❍✳ ❈♦❤♥✱ ❆✳ ❑✉♠❛r✱ ❯♥✐✈❡rs❛❧❧② ♦♣t✐♠❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣♦✐♥ts ♦♥ s♣❤❡r❡s✳ ❏♦✉r♥❛❧ ♦❢ ❆▼❙✱ ✷✵✱ ♥♦✳ ✶✱ ✾✾✲✶✹✽✱ ✷✵✵✻✳ P✳ ❇♦②✈❛❧❡♥❦♦✈✱ P✳ ❉r❛❣♥❡✈✱ ❉✳ ❍❛r❞✐♥✱ ❊✳ ❙❛✛✱ ▼✳ ❙t♦②❛♥♦✈❛✱ ❯♥✐✈❡rs❛❧ ❡♥❡r❣② ❜♦✉♥❞s ❢♦r ♣♦t❡♥t✐❛❧ ❡♥❡r❣② ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✺✵✸✳✵✼✷✷✽✮

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✻ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞s

❋♦r ✜①❡❞ str❡♥❣t❤ τ ❛♥❞ ❞✐♠❡♥s✐♦♥ n ❞❡♥♦t❡ ❜② B(n, τ) = ♠✐♥{|C| : ∃ τ✲❞❡s✐❣♥ C ⊂ Sn−✶} t❤❡ ♠✐♥✐♠✉♠ ♣♦ss✐❜❧❡ ❝❛r❞✐♥❛❧✐t② ♦❢ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥s C ⊂ Sn−✶✳ ❚❤❡♥ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞ ✐s B(n, τ) ≥ D(n, τ) =    ✷ n+k−✷

n−✶

• ,

✐❢ τ = ✷k − ✶✱ n+k−✶

n−✶

• +

n+k−✷

n−✶

• ,

✐❢ τ = ✷k.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✼ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ●❡❣❡♥❜❛✉❡r ♣♦❧②♥♦♠✐❛❧s

❋♦r ✜①❡❞ ❞✐♠❡♥s✐♦♥ n✱ t❤❡ ●❡❣❡♥❜❛✉❡r ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❞❡✜♥❡❞ ❜② P(n)

✵

= ✶, P(n)

✶

= t ❛♥❞ t❤❡ t❤r❡❡✲t❡r♠ r❡❝✉rr❡♥❝❡ r❡❧❛t✐♦♥ ✭❢♦r k ≥ ✶✮ (i + n − ✷)P(n)

i+✶(t) = (✷i + n − ✷)tP(n) i

(t) − iP(n)

i−✶(t).

■❢ f (t) ∈ R[t] ✐s ❛ r❡❛❧ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ m t❤❡♥ f (t) ❝❛♥ ❜❡ ✉♥✐q✉❡❧② ❡①♣❛♥❞❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ●❡❣❡♥❜❛✉❡r ♣♦❧②♥♦♠✐❛❧s ❛s f (t) =

m

• i=✵

fiP(n)

i

(t).

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✽ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ♠❛✐♥ ✐❞❡♥t✐t②

❚❤❡ ✐❞❡♥t✐t② |C|f (✶) +

• x,y∈C,x=y

f (x, y) = |C|✷f✵ +

m

• i=✶

fi ri

ri

• j=✶
• x∈C

vij(x) ✷ . ❤♦❧❞s tr✉❡ ❢♦r✿ ❛♥② C ⊂ ❙n−✶ ✕ ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡✱ ❛♥② f (t) = m

i=✵ fiP(n) i

(t)✱ ✇❤❡r❡ {vij(x) : j = ✶, ✷, . . . , ri} ✐s ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ t❤❡ s♣❛❝❡ ❍❛r♠(i) ♦❢ ❤♦♠♦❣❡♥❡♦✉s ❤❛r♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ i ❛♥❞ ri = ❞✐♠ ❍❛r♠(i)✳ ❆♥ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥ ♦❢ s♣❤❡r✐❝❛❧ ❞❡s✐❣♥s s❛②s t❤❛t

• x∈C

vij(x) = ✵ ❢♦r ❡✈❡r② i ≤ τ ❛♥❞ ❡✈❡r② j ≤ ri✳ ❚❤✐s s✉❣❣❡sts t❤❛t ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st τ ❝♦✉❧❞ ❜❡ ✉s❡❢✉❧ ✕ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ♠❛✐♥ ✐❞❡♥t✐t② ✐s t❤❡♥ r❡❞✉❝❡❞ t♦ |C|✷f✵✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✾ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞s ❢♦r s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✶✮

❋♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r m ✇❡ ❝♦♥s✐❞❡r t❤❡ ✐♥t❡r✈❛❧s Im =   

• t✶,✶

k−✶, t✶,✵ k

• ,

✐❢ m = ✷k − ✶,

• t✶,✵

k , t✶,✶ k

• ,

✐❢ m = ✷k. ❍❡r❡ t✶,✶

✵

= −✶✱ ta,b

i

✱ a, b ∈ {✵, ✶}✱ i ≥ ✶✱ ✐s t❤❡ ❣r❡❛t❡st ③❡r♦ ♦❢ t❤❡ ❏❛❝♦❜✐ ♣♦❧②♥♦♠✐❛❧ P

(a+ n−✸

✷ ,b+ n−✸ ✷ )

i

(t)✳ ❚❤❡ ✐♥t❡r✈❛❧s Im ❞❡✜♥❡ ♣❛rt✐t✐♦♥ ♦❢ I = [−✶, ✶) t♦ ❝♦✉♥t❛❜❧② ♠❛♥② ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❝❧♦s❡❞ s✉❜✐♥t❡r✈❛❧s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✵ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞s ❢♦r s♣❤❡r✐❝❛❧ ❝♦❞❡s ✭✷✮

❋♦r ❡✈❡r② s ∈ Im✱ ▲❡✈❡♥s❤t❡✐♥ ✉s❡❞ ❛ ♣♦❧②♥♦♠✐❛❧ f (n,s)

m

(t) ♦❢ ❞❡❣r❡❡ m ✇❤✐❝❤ s❛t✐s❢② ❛❧❧ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ❜♦✉♥❞s ❢♦r s♣❤❡r✐❝❛❧ ❝♦❞❡s✳ ❚❤✐s ②✐❡❧❞s t❤❡ ❜♦✉♥❞ A(n, s) ≤                L✷k−✶(n, s) = k+n−✸

k−✶

✷k+n−✸

n−✶

−

P(n)

k−✶(s)−P(n) k (s)

(✶−s)P(n)

k (s)

• ❢♦r s ∈ I✷k−✶,

L✷k(n, s) = k+n−✷

k

✷k+n−✶

n−✶

−

(✶+s)(P(n)

k (s)−P(n) k+✶(s))

(✶−s)(P(n)

k (s)+P(n) k+✶(s))

• ❢♦r s ∈ I✷k.

❋♦r ❡✈❡r② ✜①❡❞ ❞✐♠❡♥s✐♦♥ n ❡❛❝❤ ❜♦✉♥❞ Lm(n, s) ✐s s♠♦♦t❤ ❛♥❞ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ s✳ ❚❤❡ ❢✉♥❝t✐♦♥ L(n, s) = L✷k−✶(n, s), ✐❢ s ∈ I✷k−✶, L✷k(n, s), ✐❢ s ∈ I✷k, ✐s ❝♦♥t✐♥✉♦✉s ✐♥ s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✶ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✶✮

❚❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞ ❛♥❞ t❤❡ ▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞s ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛❧✐t✐❡s L✷k−✷(n, t✶,✶

k−✶) = L✷k−✶(n, t✶,✶ k−✶) = D(n, ✷k − ✶),

L✷k−✶(n, t✶,✵

k ) = L✷k(n, t✶,✵ k ) = D(n, ✷k)

❛♥❞ t❤❡ ❡♥❞s ♦❢ t❤❡ ✐♥t❡r✈❛❧s Im✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✷ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✷✮

❋♦r ❡✈❡r② ✜①❡❞ ✭❝❛r❞✐♥❛❧✐t②✮ N > D(n, ✷k − ✶) t❤❡r❡ ❡①✐st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ r❡❛❧ ♥✉♠❜❡rs −✶ < α✵ < α✶ < · · · < αk−✶ < ✶ ❛♥❞ ρ✵, ρ✶, . . . , ρk−✶✱ ρi > ✵ ❢♦r i = ✵, ✶, . . . , k − ✶✱ s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② f✵ = f (✶) N +

k−✶

• i=✵

ρif (αi) ❤♦❧❞s ❢♦r ❡✈❡r② r❡❛❧ ♣♦❧②♥♦♠✐❛❧ f (t) ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✷k − ✶✳ ❚❤❡ ♥✉♠❜❡rs αi✱ i = ✵, ✶, . . . , k − ✶✱ ❛r❡ t❤❡ r♦♦ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ Pk(t)Pk−✶(s) − Pk(s)Pk−✶(t) = ✵, ✇❤❡r❡ s = αk−✶✱ Pi(t) = P(n−✶)/✷,(n−✸)/✷

i

(t) ✐s ❛ ❏❛❝♦❜✐ ♣♦❧②♥♦♠✐❛❧✳ ■♥ ❢❛❝t✱ αi✱ i = ✵, ✶, . . . , k − ✶✱ ❛r❡ t❤❡ r♦♦ts ♦❢ t❤❡ ▲❡✈❡♥s❤t❡✐♥✬s ♣♦❧②♥♦♠✐❛❧ f (n,αk−✶)

✷k−✶

(t)✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✸ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✸✮

❙✐♠✐❧❛r❧②✱ ❢♦r ❡✈❡r② ✜①❡❞ ✭❝❛r❞✐♥❛❧✐t②✮ N > D(n, ✷k) t❤❡r❡ ❡①✐st ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ r❡❛❧ ♥✉♠❜❡rs −✶ = β✵ < β✶ < · · · < βk < ✶ ❛♥❞ γ✵, γ✶, . . . , γk✱ γi > ✵ ❢♦r i = ✵, ✶, . . . , k✱ s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② f✵ = f (✶) N +

k

• i=✵

γif (βi) ✭✶✮ ✐s tr✉❡ ❢♦r ❡✈❡r② r❡❛❧ ♣♦❧②♥♦♠✐❛❧ f (t) ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✷k✳ ❚❤❡ ♥✉♠❜❡rs βi✱ i = ✵, ✶, . . . , k✱ ❛r❡ t❤❡ r♦♦ts ♦❢ t❤❡ ▲❡✈❡♥s❤t❡✐♥✬s ♣♦❧②♥♦♠✐❛❧ f (n,βk)

✷e

(t)✳ ❱✳ ■✳ ▲❡✈❡♥s❤t❡✐♥✱ ❉❡s✐❣♥s ❛s ♠❛①✐♠✉♠ ❝♦❞❡s ✐♥ ♣♦❧②♥♦♠✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❆❝t❛ ❆♣♣❧✳ ▼❛t❤✳ ✷✺✱ ✶✾✾✷✱ ✶✲✽✷✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✹ ✴ ✸✽

Pr❡❧✐♠✐♥❛r✐❡s ✕ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❉●❙✲ ❛♥❞ ▲✲❜♦✉♥❞s ✭✹✮

■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ❛❧✇❛②s t❛❦❡ ❝❛r❡ ✇❤❡r❡ t❤❡ ❝❛r❞✐♥❛❧✐t② N ✐s ❧♦❝❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❉❡❧s❛rt❡✲●♦❡t❤❛❧s✲❙❡✐❞❡❧ ❜♦✉♥❞✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❜♦✉♥❞s D(n, τ) ❛♥❞ Lm(n, s) t❤❛t N ∈ [D(n, τ), D(n, τ + ✶)] ⇐ ⇒ s ∈ Im, ✇❤❡r❡ s ❛♥❞ N ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② t❤❡ ❡q✉❛❧✐t② N = Lτ(n, s). ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ❛❧✇❛②s ❛ss♦❝✐❛t❡ N ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♥✉♠❜❡rs✿ α✵, α✶, . . . , αk−✶, ρ✵, ρ✶, . . . , ρk−✶ ✇❤❡♥ N ∈ [D(n, ✷k − ✶), D(n, ✷k)) ♦r ✇✐t❤ β✵, β✶, . . . , βk, γ✵, γ✶, . . . , γk ✇❤❡♥ N ∈ [D(n, ✷k), D(n, ✷k + ✶)).

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✺ ✴ ✸✽

▲P ❜♦✉♥❞s ✕ ❧♦✇❡r ❜♦✉♥❞s ❢♦r L(n, N, τ; h)

❚❤❡♦r❡♠ ✶✳ ▲❡t N✱ n✱ τ ❛♥❞ h ❜❡ ✜①❡❞ ❛♥❞ f (t) ❜❡ ❛ r❡❛❧ ♣♦❧②♥♦♠✐❛❧ s✉❝❤ t❤❛t ✭❆✶✮ f (t) ≤ h(t) ❢♦r −✶ ≤ t ≤ ✶✳ ✭❆✷✮ t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ●❡❣❡♥❜❛✉❡r ❡①♣❛♥s✐♦♥ f (t) = ❞❡❣(f )

i=✵

fiP(n)

i

(t) s❛t✐s❢② fi ≥ ✵ ❢♦r i ≥ τ + ✶✳ ❚❤❡♥ L(n, N, τ; h) ≥ N(f✵N − f (✶))✳ An,N,τ;h ✕ t❤❡ s❡t ♦❢ ❣♦♦❞ ♣♦❧②♥♦♠✐❛❧s

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✻ ✴ ✸✽

▲P ❜♦✉♥❞s ✕ ✉♣♣❡r ❜♦✉♥❞s ❢♦r U(n, N, τ; h)

❚❤❡♦r❡♠ ✷✳ ▲❡t N✱ n✱ τ ❛♥❞ h ❜❡ ✜①❡❞✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts t✵ ∈ [−✶, ✶] s✉❝❤ t❤❛t ♥♦ τ✲❞❡s✐❣♥ ♦♥ Sn−✶ ♦❢ N ♣♦✐♥ts ❝❛♥ ❤❛✈❡ ✐♥♥❡r ♣r♦❞✉❝ts ✐♥ t❤❡ ✐♥t❡r✈❛❧ (t✵, ✶)✳ ▲❡t g(t) ❜❡ ❛ r❡❛❧ ♣♦❧②♥♦♠✐❛❧ s✉❝❤ t❤❛t ✭❇✶✮ g(t) ≥ h(t) ❢♦r ❡✈❡r② t ∈ [−✶, t✵]✱ ✭❇✷✮ t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ●❡❣❡♥❜❛✉❡r ❡①♣❛♥s✐♦♥ g(t) = ❞❡❣(g)

i=✵

giP(n)

i

(t) s❛t✐s❢② gi ≤ ✵ ❢♦r i ≥ τ + ✶✳ ❚❤❡♥ U(n, N, τ; h) ≤ N(g✵N − g(✶))✳ Bn,N,τ;h ✕ t❤❡ s❡t ♦❢ ❣♦♦❞ ♣♦❧②♥♦♠✐❛❧s

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✼ ✴ ✸✽

▲♦✇❡r ❜♦✉♥❞s ✭❯▲❇✮ ✭✶✮

❲❡ ♥❡❡❞ ❍❡r♠✐t❡✬s ✐♥t❡r♣♦❧❛t✐♦♥ t♦ h(t) ❛s ❢♦❧❧♦✇s✳ ❉❡✜♥❡ ✭✐✮ t❤❡ ♣♦❧②♥♦♠✐❛❧ f (t) ♦❢ ❞❡❣r❡❡ τ = ✷k − ✶ ❜② f (αi) = h(αi), f ′(αi) = h′(αi), i = ✵, ✶, . . . , k − ✶. ✭✐✐✮ t❤❡ ♣♦❧②♥♦♠✐❛❧ f (t) ♦❢ ❞❡❣r❡❡ τ = ✷k ❜② f (β✵) = h(β✵), f (βi) = h(βi), f ′(βi) = h′(βi), i = ✶, . . . , k. ❚❤❡s❡ ❝♦♥❞✐t✐♦♥s ❞❡✜♥❡ ❛ ❍❡r♠✐t❡✬s ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r f (t) t♦ ✐♥t❡rs❡❝t ❛♥❞ t♦✉❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ h(t)✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ✭❆✶✮ ✐s s❛t✐s✜❡❞✳ ▼♦r❡♦✈❡r✱ ❞❡❣(f ) = τ ❛♥❞ ✭❆✷✮ ✐s ✭tr✐✈✐❛❧❧②✮ s❛t✐s✜❡❞✳ ❚❤✉s ✇❡ ❝❛♥ ✉s❡ f (t) ❢♦r ❜♦✉♥❞✐♥❣ L(n, N, τ; h) ❢r♦♠ ❜❡❧♦✇✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✽ ✴ ✸✽

▲♦✇❡r ❜♦✉♥❞s ✭❯▲❇✮ ✭✷✮

❚❤❡♦r❡♠ ✸✳ ▲❡t n✱ τ✱ N ∈ (D(n, τ), D(n, τ + ✶)] ❛♥❞ h ❜❡ ✜①❡❞✳ ❚❤❡♥ t❤❡ ♣♦❧②♥♦♠✐❛❧s ❢r♦♠ ✭✐✮ ❛♥❞ ✭✐✐✮ ❣✐✈❡ t❤❡ ❜♦✉♥❞s L(n, N, ✷k − ✶; h) ≥ N✷

k−✶

• i=✵

ρih(αi), L(n, N, ✷k; h) ≥ N✷

k

• i=✵

γih(βi), r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡s❡ ❜♦✉♥❞s ❝❛♥ ♥♦t ❜❡ ✐♠♣r♦✈❡❞ ❜② ✉s✐♥❣ ♣♦❧②♥♦♠✐❛❧s ❢r♦♠ An,N,✷k−✶;h ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✷k − ✶ ❛♥❞ An,N,✷k;h ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✷k✱ r❡s♣❡❝t✐✈❡❧②✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✶✾ ✴ ✸✽

❚✇♦ ✇❛②s ❢♦r ✐♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❙❤♦rt❡r ✐♥t❡r✈❛❧s ❛♥❞ ❍✐❣❤❡r ❞❡❣r❡❡s

❚❤❡ ❯▲❇ ❜♦✉♥❞s ❛r❡ ♦♣t✐♠❛❧ ✐♥ s♦♠❡ s❡♥s❡ ✕ t❤❡② ❝❛♥ ♥♦t ❜❡ ✐♠♣r♦✈❡❞ ❜② ♣♦❧②♥♦♠✐❛❧s ❢r♦♠ An,M,τ;h ♦❢ ❞❡❣r❡❡ τ ♦r ❧♦✇❡r✳ ❋✐rst ✇❛② ❢♦r ♦❜t❛✐♥✐♥❣ ❜❡tt❡r ❜♦✉♥❞s ✕ ♠❛❦✐♥❣ ❜❡tt❡r ▲P ❜② s✉❜✐♥t❡r✈❛❧s ♦❢ [−✶, ✶) ❜❛s❡❞ ♦♥ ♣r❡❧✐♠✐♥❛r② ✭♥♦♥tr✐✈✐❛❧✮ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ✐♥♥❡r ♣r♦❞✉❝ts ✭♦❢ τ✲❞❡s✐❣♥s ♦❢ N ♣♦✐♥ts ♦♥ Sn−✶✮✳ ❚❤✐s ✐s ❡①❛❝t❧② t❤❡ ❝❛s❡ ✇❤❡♥ τ ✐s ❡✈❡♥✳ ❙❡❝♦♥❞ ✇❛② ❢♦r ♦❜t❛✐♥✐♥❣ ❜❡tt❡r ❜♦✉♥❞s ✕ ✉s✐♥❣ ▲P ✇✐t❤ ❤✐❣❤❡r ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡r❡ ❛r❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ♦❢ ❯▲❇✱ ❛♥❞ ✇❡ ❝❛♥ ❞♦ ❜❡tt❡r ✇❤❡♥ t❤❡ ❯▲❇ ✐♥ ♥♦t ❣❧♦❜❛❧❧② ♦♣t✐♠❛❧✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✵ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❙❤♦rt❡r ✐♥t❡r✈❛❧ ✭✶✮

❉❡♥♦t❡ u(n, N, τ) = s✉♣{u(C) : C ⊂ Sn−✶ ✐s ❛ τ✲❞❡s✐❣♥, |C| = N}, ✇❤❡r❡ u(C) = ♠❛①{x, y : x, y ∈ C, x = y}✱ ❛♥❞ ℓ(n, N, τ) = ✐♥❢{ℓ(C) : C ⊂ Sn−✶ ✐s ❛ τ✲❞❡s✐❣♥, |C| = N}, ✇❤❡r❡ ℓ(C) = ♠✐♥{x, y : x, y ∈ C, x = y}✳ ❋♦r ❡✈❡r② n✱ τ ❛♥❞ ❝❛r❞✐♥❛❧✐t② N ∈ [D(n, τ), D(n, τ + ✶)] ♥♦♥✲tr✐✈✐❛❧ ✉♣♣❡r ❜♦✉♥❞s ♦♥ u(n, N, τ) ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✳ ❙✐♠✐❧❛r❧②✱ ❢♦r ❡✈❡♥ τ = ✷k ❛♥❞ ❝❛r❞✐♥❛❧✐t② N ∈ [D(n, ✷k), D(n, ✷k + ✶)) ♥♦♥✲tr✐✈✐❛❧ ❧♦✇❡r ❜♦✉♥❞s ♦♥ ℓ(n, N, ✷k) ❛r❡ ♣♦ss✐❜❧❡✳ ❲❡ ❞❡s❝r✐❜❡ ❤❡r❡ ❡①♣❧✐❝✐t❧② t❤❡ ❝❛s❡s τ = ✷ ❛♥❞ τ = ✹✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✶ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❙❤♦rt❡r ✐♥t❡r✈❛❧ ✭✷✮

❋✉rt❤❡r ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥✿ ❆ s♣❤❡r✐❝❛❧ τ✲❞❡s✐❣♥ C ⊂ Sn−✶ ✐s ❛ s♣❤❡r✐❝❛❧ ❝♦❞❡ s✉❝❤ t❤❛t

• y∈C

f (x, y) = f✵|C|. ❤♦❧❞s ❢♦r ❛♥② ♣♦✐♥t x ∈ Sn−✶ ❛♥❞ ❛♥② r❡❛❧ ♣♦❧②♥♦♠✐❛❧ f (t) = r

i=✵ fiP(n) i

(t) ♦❢ ❞❡❣r❡❡ ❛t ♠♦st τ✳ ▲❡♠♠❛✳ ❛✮ ❋♦r ❡✈❡r② n ≥ ✸ ❛♥❞ ❡✈❡r② N ∈ [D(n, ✷), D(n, ✸)] = [n + ✶, ✷n] ✇❡ ❤❛✈❡ u(n, N, ✷) ≤ N − ✷ n − ✶. ❜✮ ❋♦r ❡✈❡r② n ≥ ✸ ❛♥❞ ❡✈❡r② N ∈ [D(n, ✹), D(n, ✺)] = [n(n + ✸)/✷, n(n + ✶)] ✇❡ ❤❛✈❡ u(n, N, ✹) ≤ ✷(✸ +

• (n − ✶)[(n + ✷)N − ✸(n + ✸)])

n(n + ✷) − ✶.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✷ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❙❤♦rt❡r ✐♥t❡r✈❛❧ ✭✸✮

▲❡♠♠❛✳ ❛✮ ❋♦r ❡✈❡r② n ≥ ✸ ❛♥❞ ❡✈❡r② N ∈ [D(n, ✷), D(n, ✸)] = [n + ✶, ✷n] ✇❡ ❤❛✈❡ ℓ(n, N, ✷) ≥ ✶ − N n . ❜✮ ❋♦r ❡✈❡r② n ≥ ✸ ❛♥❞ ❢♦r ❡✈❡r② N ∈ [D(n, ✹), D(n, ✺)] = [n(n + ✸)/✷, n(n + ✶)] ✇❡ ❤❛✈❡ ℓ(n, N, ✹) ≥ ✶ − ✷ n

• ✶ +
• (n − ✶)(N − ✷)

n + ✷

• .

❋✉rt❤❡r ❜♦✉♥❞s ♦♥ u(n, N, τ) ❛♥❞ ℓ(n, N, τ) ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❛ t❡❝❤♥✐q✉❡ ❢r♦♠ ❇✳✲❇♦✉♠♦✈❛✲❉❛♥❡✈✱ ◆❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ❡①✐st❡♥❝❡ ♦❢ s♦♠❡ ❞❡s✐❣♥s ✐♥ ♣♦❧②♥♦♠✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❊✉r♦♣✳ ❏✳ ❈♦♠❜✐♥✳✱ ✷✵ ✷✶✸✲✷✷✺✱ ✶✾✾✾✳ ❋♦r τ ≥ ✹ s✉❝❤ ❜♦✉♥❞s ❛r❡ ❜❡tt❡r ✐♥ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s t❤❛♥ t❤❡s❡ ❢r♦♠ t❤❡ ▲❡♠♠❛s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✸ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❙❤♦rt❡r ✐♥t❡r✈❛❧ ✭✹✮

❚❤❡♦r❡♠✳ ▲❡t n✱ N ∈ [D(n, ✷k), D(n, ✷k + ✶)]✱ τ = ✷k ❛♥❞ h ❜❡ ✜①❡❞✳ ▲❡t f (t) ❜❡ ❛ r❡❛❧ ♣♦❧②♥♦♠✐❛❧ ✇❤✐❝❤ s❛t✐s✜❡s ✭❆✷✮ ❛♥❞ ✭❆✶′✮ f (t) ≤ h(t) ❢♦r ℓ(n, N, ✷k) ≤ t ≤ u(n, N, ✷k)✳ ❚❤❡♥ L(n, N, ✷k; h) ≥ N(f✵N − f (✶))✳ ❚❤❡♦r❡♠✳ ✭❞❡❣r❡❡ t✇♦✮ ❋♦r ❡✈❡r② n✱ N ∈ [n + ✶, ✷n] ❛♥❞ h✿ L(n, N, ✷; h) ≥ N[h(✵)N(N − n − ✶) + nh(✶ − N/n)] N − n . ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳ ❯s❡ f (t)✿ f (ℓ) = h(ℓ)✱ f (a) = h(a) ❛♥❞ f ′(a) = h′(a) ❢♦r s♦♠❡ a ∈ (ℓ, ✶)✳ ❚❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ a t♦ ♠❛①✐♠✐③❡ f✵N − f (✶) ❣✐✈❡s ❜❡st ✈❛❧✉❡ ❛t a✵ =

n(✶−ℓ)−N n(✶−ℓ)+ℓNn ✇❤✐❝❤ t✉r♥s t♦ a✵ = ✵ ❢♦r

ℓ = ✶ − N/n✳ P❧✉❣ ✐♥ f✵N − f (✶) t♦ ❣❡t t❤❡ ❞❡s✐r❡❞ ❜♦✉♥❞✳

• P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙

❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✹ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❙❤♦rt❡r ✐♥t❡r✈❛❧ ✭✺✮

❈♦r♦❧❧❛r②✳ ■❢ n ❛♥❞ N = ξn✱ ξ ∈ (✶, ✷) ✐s ❝♦♥st❛♥t✱ t❡♥❞ s✐♠✉❧t❛♥❡♦✉s❧② t♦ ✐♥✜♥✐t② t❤❡♥ L(n, N, ✷; h) h(✵)N✷ + N[h(✶ − ξ) − ξh(✵)] ξ − ✶ . Pr♦♦❢✳ P❧✉❣ n = N/ξ ❛♥❞ ℓ = ✶ − ξ✳

• ▲♦✇❡r ❜♦✉♥❞s ❢♦r t❤❡ ❡♥❡r❣② ♦❢ ✹✲❞❡s✐❣♥s ✐♥ s❤♦rt❡r ✐♥t❡r✈❛❧s ❝❛♥ ❜❡

♦❜t❛✐♥❡❞ ❜② ✐♥t❡r♣♦❧❛t✐♦♥ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❢♦✉r✿ f (ℓ) = h(ℓ), f (a) = h(a), f ′(a) = h′(a), f (b) = h(b), f ′(b) = h′(b), ✇❤❡r❡ t❤❡ t♦✉❝❤✐♥❣ ♣♦✐♥ts a ❛♥❞ b ♠✉st ❜❡ ❝❤♦s❡♥ t♦ ♠❛①✐♠✐③❡ f✵N − f (✶) ✕ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s t❛❧❦✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✺ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❍✐❣❤❡r ❞❡❣r❡❡s ✭✶✮

▲❡t n ❛♥❞ N ❜❡ ✜①❡❞✱ N ∈ [D(n, ✷k − ✶), D(n, ✷k))✱ Lτ(n, s) = N ❛♥❞ j ❜❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s ✐♥ n ❛♥❞ s ∈ I✷k−✶ Qj(n, s) = ✶ N +

k−✶

• i=✵

ρiP(n)

j

(αi) ✭✷✮ ✭♥♦t❡ t❤❛t P(n)

j

(✶) = ✶✮✳ ■t ❢♦❧❧♦✇s t❤❛t Qj(n, s) = ✵ ❢♦r ❡✈❡r② ✶ ≤ j ≤ ✷k − ✶ ❛♥❞ ❡✈❡r② s ∈ I✷k−✶ ✭s✐♥❝❡ t❤✐s ✐s t❤❡ ❝♦❡✣❝✐❡♥t f✵ = ✵ ✐♥ t❤❡ ●❡❣❡♥❜❛✉❡r ❡①♣❛♥s✐♦♥ ♦❢ P(n)

j

(t)✮✳ ❙♦ t❤❡ ❢✉♥❝t✐♦♥s Qj(n, s) ❛r❡ ♥♦t ✐♥t❡r❡st✐♥❣ ✐♥ t❤❡s❡ ❝❛s❡s ❛♥❞ ✇❡ ❛ss✉♠❡ ❜❡❧♦✇ t❤❛t j ≥ ✷k✳ ❚❤❡ ♥❡①t t❤❡♦r❡♠ s❤♦✇s t❤❛t t❤❡ ❢✉♥❝t✐♦♥s Qj(n, s) ❣✐✈❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r ❡①✐st❡♥❝❡ ♦❢ ✐♠♣r♦✈✐♥❣ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❤✐❣❤❡r ❞❡❣r❡❡s✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✻ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❍✐❣❤❡r ❞❡❣r❡❡s ✭✷✮

❚❤❡♦r❡♠✳ ❆ss✉♠❡ t❤❛t h ✐s ❝♦♠♣❧❡t❡❧② ♠♦♥♦t♦♥❡✳ ❚❤❡♥ t❤❡ ❜♦✉♥❞ L(n, N, ✷k − ✶; h) ≥ N✷

k−✶

• i=✵

ρih(αi) ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ❢r♦♠ An,N,✷k−✶;h ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st ✷k ✐❢ ❛♥❞ ♦♥❧② ✐❢ Qj(n, s) < ✵ ❢♦r s♦♠❡ j ≥ ✷k✳ ▼♦r❡♦✈❡r✱ ✐❢ Qj(n, s) < ✵ ❢♦r s♦♠❡ j ≥ ✷k✱ t❤❡♥ t❤❛t ❜♦✉♥❞ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ❢r♦♠ An,N,✷k−✶;h ♦❢ ❞❡❣r❡❡ ❡①❛❝t❧② j✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✼ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❍✐❣❤❡r ❞❡❣r❡❡s ✭✸✮

❚❤❡ t❡st ❢✉♥❝t✐♦♥s Qj(n, s) ❝♦✐♥❝✐❞❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡ s❛♠❡ ♥❛♠❡ ✇❤✐❝❤ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❛♥❞ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❇✳✲❉❛♥❡✈✲❇✉♠♦✈❛✱ ❯♣♣❡r ❜♦✉♥❞s ♦♥ t❤❡ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ♦❢ s♣❤❡r✐❝❛❧ ❝♦❞❡s✱ ■❊❊❊ ❚r❛♥s✳ ■♥❢♦r♠✳ ❚❤❡♦r② ✹✷✱ ✶✾✾✻✱ ✶✺✼✻✲✶✺✽✶✳ ❚❤❡♦r❡♠✳ ❚❤❡ ❜♦✉♥❞s L(n, N, ✷k − ✶; h) ≥ N✷ k−✶

i=✵ ρih(αi) ❝❛♥ ♥♦t

❜❡ ✐♠♣r♦✈❡❞ ❜② ✉s✐♥❣ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡s ✷k ❛♥❞ ✷k + ✶✳ ❈♦r♦❧❧❛r②✳ ❆♥② ✐♠♣r♦✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ♠✉st ❤❛✈❡ ❞❡❣r❡❡ ❛t ❧❡❛st ✷k + ✷✳ ❆❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣ ✐♠♣r♦✈✐♥❣ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡s ✷k + ✷ ❛♥❞ ✷k + ✸ ✕ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s t❛❧❦✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✽ ✴ ✸✽

■♠♣r♦✈✐♥❣ ❯▲❇ ✕ ❍✐❣❤❡r ❞❡❣r❡❡s ✭✹✮

❚❤❡♦r❡♠✳ ❚❤❡ ❢✉♥❝t✐♦♥ Q✷k+✸(n, s) ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❢♦r n ≥ ✸✱ k ≥ ✷ ❛♥❞ s ∈ I✷k−✶✿ ■❢ k ≥ ✾ ❛♥❞ ✸ ≤ n ≤ k✷−✹k+✺+

√ k✹−✽k✸−✻k✷+✷✹k+✷✺ ✹

✱ t❤❡♥ Q✷k+✸(n, s) < ✵ ❢♦r ❡✈❡r② s ∈

• t✶,✶

e−✶, t✶,✵ e

• ✳

❚❤✉s ✐♥ ✜①❡❞ ❞✐♠❡♥s✐♦♥ ❛❧❧ ❯▲❇s L(n, N, ✷k − ✶; h) ≥ N✷ k−✶

i=✵ ρih(αi) ✇✐t❤ s✉✣❝✐❡♥t❧② ❧❛r❣❡ k ❝❛♥ ❜❡

✐♠♣r♦✈❡❞✳ ❈♦r♦❧❧❛r②✳ ❚❤❡r❡ ❝♦✉❧❞ ❡①✐st ♦♥❧② ✜♥✐t❡❧② ♠❛♥② s❤❛r♣ ❝♦♥✜❣✉r❛t✐♦♥s ✭t❤❡ s❛♠❡ ❛s ❝♦❞❡s ❛tt❛✐♥✐♥❣ t❤❡ ♦❞❞ ▲❡✈❡♥s❤t❡✐♥ ❜♦✉♥❞ L✷k−✶(n, s) ❢♦r s♦♠❡ n ≥ ✸ ❛♥❞ s ∈ I✷k−✶ ❛♥❞ k ≥ ✷✮ ✐♥ ❛♥② ✜①❡❞ ❞✐♠❡♥s✐♦♥✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✷✾ ✴ ✸✽

❯♣♣❡r ❜♦✉♥❞s ✭✶✮

❙✉✐t❛❜❧❡ t✵ ❢♦r ❚❤❡♦r❡♠ ✷ ❛r❡ t❤❡ ♥♦♥tr✐✈✐❛❧ ✉♣♣❡r ❜♦✉♥❞s ♦♥ u(n, N, τ)✳ ❲❡ ❞♦ ♥♦t ❤❛✈❡ ❣❡♥❡r❛❧ ✭✉♥✐✈❡rs❛❧✮ ✉♣♣❡r ❜♦✉♥❞s s♦ ❢❛r ❛♥❞ s❤♦✇ ❡①❛♠♣❧❡s ❢♦r s♠❛❧❧ ❞❡❣r❡❡s ✭✉♣ t♦ ❢♦✉r✮✳ ❯♣♣❡r ❜♦✉♥❞ ❢♦r ✷✲❞❡s✐❣♥s✿ U(n, N, ✷; h) ≤ N[(N − ✶)(uh(ℓ) − ℓh(u)) + h(ℓ) − h(u)] u − ℓ ❢♦r ❡✈❡r② n✱ N ∈ [n + ✶, ✷n] ❛♥❞ h✳ Pr♦♦❢✳ ❯s❡ t❤❡ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✇❤✐❝❤ ❣r❛♣❤ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts (ℓ, h(ℓ)) ❛♥❞ (u, h(u))✳

• P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙

❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✵ ✴ ✸✽

❯♣♣❡r ❜♦✉♥❞s ✭✷✮

■❢ n ❛♥❞ N = ξn✱ ξ ∈ (✶, ✷) ✐s ❝♦♥st❛♥t✱ t❡♥❞ s✐♠✉❧t❛♥❡♦✉s❧② t♦ ✐♥✜♥✐t②✱ t❤❡♥ U(n, N, ✷; h) N✷ h(✶ − ξ) + h(ξ − ✶) ✷ + (✷ − ξ)h(✶ − ξ) − ξh(ξ − ✶) ✷N(ξ − ✶) . ❲❡ ♥♦✇ ❤❛✈❡ ❛♥ ❛s②♠♣t♦t✐❝ str✐♣ ❢♦r t❤❡ ❡♥❡r❣② E ♦❢ s♣❤❡r✐❝❛❧ ✷✲❞❡s✐❣♥s ♦❢ N = ξn ∈ [n + ✶, ✷n − ✶] ♣♦✐♥ts✿ h(✵) + h(✶ − ξ) − ξh(✵) (✶ − ξ)N E N✷

• h(✶ − ξ) + h(ξ − ✶)

✷ + (✷ − ξ)h(✶ − ξ) − ξh(ξ − ✶) ✷(ξ − ✶)N .

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✶ ✴ ✸✽

❯♣♣❡r ❜♦✉♥❞s ✭✸✮

❯♣♣❡r ❜♦✉♥❞ ❢♦r ✸✲ ❛♥❞ ✹✲❞❡s✐❣♥s✿ U(n, N, τ; h) ≤ N(N − ✶)h(a✵) + + (h(ℓ) − h(a✵))

• uN(✶ + na✷

✵) + ✷Na✵ + n(✶ − u)(✶ − a✵)✷

n(u − ℓ)(ℓ − a✵)✷ + (h(u) − h(a✵))

• ℓN(✶ + na✷

✵) + ✷Na✵ + n(✶ − ℓ)(✶ − a✵)✷

n(u − ℓ)(u − a✵)✷ , ❢♦r τ = ✸✱ ❡✈❡r② n ❛♥❞ N ∈ [✷n, n(n+✸)

✷

]✱ ❛♥❞ ❢♦r τ = ✹✱ ❡✈❡r② n ❛♥❞ N ∈ [ n(n+✸)

✷

, n✷ + n]✱ ✇❤❡r❡ a✵ =

N(ℓ+u)+n(✶−ℓ)(✶−u) n(✶−ℓ)(✶−u)−N(✶+ℓun)✳

Pr♦♦❢✳ ❯s❡ t❤✐r❞ ❞❡❣r❡❡ ♣♦❧②♥♦♠✐❛❧ ✇❤✐❝❤ ❣r❛♣❤ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣♦✐♥ts (ℓ, h(ℓ)) ❛♥❞ (u, h(u)) ❛♥❞ t♦✉❝❤❡s t❤❡ ❣r❛♣❤ ♦❢ h(t) ✭❢r♦♠ ❛❜♦✈❡✮ ❛t t❤❡ ♣♦✐♥t (a✵, h(a✵))✳

• P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙

❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✷ ✴ ✸✽

❯♣♣❡r ❜♦✉♥❞s ✭✹✮

❚❤❡♦r❡♠✳ ■❢ n ❛♥❞ N t❡♥❞ t♦ ✐♥✜♥✐t② ✐♥ r❡❧❛t✐♦♥ N = n✷ξ✱ ✇❤❡r❡ ξ ∈ [✶/✷, ✶) ✐s ❛ ❝♦♥st❛♥t✱ t❤❡♥ U(n, N, ✹; h) h(✵)N✷ − h(✵)N + c✶ √ N + c✷, ✇❤❡r❡ c✶ ❛♥❞ c✷ ❛r❡ ❝❡rt❛✐♥ ❝♦♥st❛♥ts✳ Pr♦♦❢✳ ❚❤❡ ❛s②♠♣t♦t✐❝ ❢♦r♠s ♦❢ ♦✉r ♣❛r❛♠❡t❡rs ✐s✿ u(n, N, ✹) ∼ ✷

• ξ − ✶ ✭❢r♦♠ ▲❡♠♠❛✮,

ℓ(n, N, ✹) ∼ ✶ − ✷

• ξ ✭❢r♦♠ ▲❡♠♠❛✮,

a✵ ∼ ✵ ✭❢r♦♠ ❛❜♦✈❡✮, ♥♦✇ ♣❧✉❣ t❤❡s❡✳

• P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙

❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✸ ✴ ✸✽

❊①❛♠♣❧❡✿ ❝♦♠♣❛r❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ❢♦r ✷✲❞❡s✐❣♥s

❚❤❡ ✉♣♣❡r ❜♦✉♥❞s ❝♦✐♥❝✐❞❡ ❡①❛❝t❧② ✇❤❡♥ N = n + ✶ ♦r N = n + ✷ ❢♦r ❡✈❡r② n ❛♥❞ h✳ ❚❤❡ ❝❛s❡ N = n + ✶ ❧❡❛❞s t♦ t❤❡ r❡❣✉❧❛r s✐♠♣❧❡① ♦♥ Sn−✶✳ ❚❤❡ ❝❛s❡ N = n + ✷ ✐s ♠♦r❡ ✐♥t❡r❡st✐♥❣ ✕ ▼✐♠✉r❛ ✭✶✾✾✵✮ ❤❛s ♣r♦✈❡❞ t❤❛t s♣❤❡r✐❝❛❧ ✷✲❞❡s✐❣♥s ✇✐t❤ n + ✷ ♣♦✐♥ts ♦♥ Sn−✶ ❞♦ ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢ n ✐s ❡✈❡♥✳ ❚❤❡ ♥♦♥❡①✐st❡♥❝❡ r❡s✉❧t ❢♦❧❧♦✇s ❡❛s✐❧② ❢r♦♠ t❤❡ ❝♦✐♥❝✐❞❡♥❝❡✳ ■t ❛❧s♦ ❢♦❧❧♦✇s t❤❛t t❤❡ ✷✲❞❡s✐❣♥s ♦❢ n + ✷ ♣♦✐♥ts ❢♦r ❡✈❡♥ n ❛r❡ ✉♥✐q✉❡ ✭✜rst❧② ♣r♦✈❡❞ ❜② ❙❛❧✐ ✐♥ ✶✾✾✸✮ ❛♥❞ ♦♣t✐♠❛❧ ✕ t❤❡② ❤❛✈❡ s✐♠✉❧t❛♥❡♦✉s❧② ♠✐♥✐♠✉♠ ❛♥❞ ♠❛①✐♠✉♠ ♣♦ss✐❜❧❡ ❡♥❡r❣②✳

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✹ ✴ ✸✽

❆s②♠♣t♦t✐❝ ❜♦✉♥❞s ✭✶✮

▲❡t t❤❡ ❞✐♠❡♥s✐♦♥ n ❛♥❞ t❤❡ ❝❛r❞✐♥❛❧✐t② N t❡♥❞ s✐♠✉❧t❛♥❡♦✉s❧② t♦ ✐♥✜♥✐t② ✐♥ t❤❡ r❡❧❛t✐♦♥ ❧✐♠ N nk−✶ = ✶ (k − ✶)! + γ, ✇❤❡r❡ γ ≥ ✵ ✐s ❛ ❝♦♥st❛♥t✱ ✐✳❡✳ N ∼ nk−✶(

✶ (k−✶)! + γ)✳

❲❡ ❦♥♦✇ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✿ αi ∼ ✵, ❢♦r i = ✶, ✷, . . . , k − ✶, α✵ ∼ − ✶ ✶ + γ(k − ✶)!, ρ✵N ∼ (✶ + γ(k − ✶)!)✷k−✶.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✺ ✴ ✸✽

❆♣♣❧✐❝❛t✐♦♥s ✕ ❛s②♠♣t♦t✐❝ ❜♦✉♥❞s ✭✷✮

◆♦✇ t❤❡ ❜♦✉♥❞s ❛r❡ ❡❛s② t♦ ❜❡ ❝❛❧❝✉❧❛t❡❞ ✕ L(n, N, ✷k − ✶; h) ≥ N✷

k−✶

• i=✵

ρih(αi) ∼ h(✵)N✷. ❚❤❡ ❜♦✉♥❞ ♦❢ h(✵)N✷ ∼ ch(✵)n✹ ✐s ❛tt❛✐♥❡❞ ❜② t❤❡ s♣❤❡r✐❝❛❧ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❑❡r❞♦❝❦ ❝♦❞❡s ✭✐♥ ❞✐♠❡♥s✐♦♥s ✷✷ℓ✮✳ ❙✐♠✐❧❛r❧②✱ ✐♥ t❤❡ ❡✈❡♥ ❝❛s❡ ✇❡ ♦❜t❛✐♥ L(n, N, ✷k; h) h(✵)N✷.

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✻ ✴ ✸✽

❋✉t✉r❡ ✇♦r❦

• ❡♥❡r❛❧ ✭✉♥✐✈❡rs❛❧✮ ❜♦✉♥❞s ✐♥ s❤♦rt❡r ✐♥t❡r✈❛❧s
• ❡♥❡r❛❧ ✭✉♥✐✈❡rs❛❧✮ ❜♦✉♥❞s ❜② ❤✐❣❤❡r ❞❡❣r❡❡s
• ❡♥❡r❛❧ ✭✉♥✐✈❡rs❛❧✮ ✉♣♣❡r ❜♦✉♥❞s

❇♦✉♥❞s ❢♦r ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ✐♥ ❍❛♠♠✐♥❣ s♣❛❝❡s✱ ✇✐t❤ s♣❡❝✐❛❧ ✐♥t❡r❡st t♦ t❤❡ ❜✐♥❛r② ❝❛s❡ ❇♦✉♥❞s ❢♦r ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ✐♥ ❏♦❤♥s♦♥ s♣❛❝❡s ❇♦✉♥❞s ❢♦r ❝♦❞❡s ❛♥❞ ❞❡s✐❣♥s ✐♥ ✐♥✜♥✐t❡ ♣r♦❥❡❝t✐✈❡ s♣❛❝❡s ❇♦✉♥❞s ❢♦r ❡♥❡r❣② ♦❢ ❊✉❝❧✐❞❡❛♥ ❞❡s✐❣♥s

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✼ ✴ ✸✽

❚❍❆◆❑ ❨❖❯ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆ ✦

P❇✱ P❉✱ ❉❍✱ ❊❙✱ ▼❙ ❯♥✐✈❡rs❛❧ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ♦♥ ❡♥❡r❣② ♦❢ ❞❡s✐❣♥s ✐♥ Sn−✶ ❛♥❞ H(n, q) ✷✶✲✷✹✳✵✹✳✷✵✶✺ ✸✽ ✴ ✸✽