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❉❡♥❥♦② ❡①❛♠♣❧❡s ❛♥❞ t❤❡✐r ❞✐♠❡♥s✐♦♥

❘✉❞✐♠❡♥t❛r② s❧✐❞❡s

❾✉❦❛s③ P❛✇❡❧❡❝

❙●❍ ❲❛rs❛✇ ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s

❲❛rs③❛✇❛✱ ❆♣r✐❧ ✷✵✷✵

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇✐❜❧✐♦❣r❛♣❤②

❚❤✐s t❛❧❦ ✐s ♣❛rt✐❛❧❧② ❜❛s❡❞ ♦♥✿ ❉✐♦♣❤❛♥t✐♥❡ ❝❧❛ss❡s✱ ❞✐♠❡♥s✐♦♥ ❛♥❞ ❉❡♥❥♦② ♠❛♣s ❜② ❇✳ ❑r❛ ❛♥❞ ❏✳ ❙❝❤♠❡❧❧✐♥❣✱ ❆❝t❛ ❆r✐t❤♠❡t✐❝❛ ✶✵✺✳✹ ✭✷✵✵✷✮❀ ❲♦r❦ ✐♥ ♣r♦❣r❡ss ✇✐t❤ ▼✳ ❯r❜❛➠s❦✐❀ ❙♦♠❡ r❡s✉❧ts ❢r♦♠ ♠② P❤❉✱ s♦♠❡❞❛② ♣❡r❤❛♣s ♣✉❜❧✐s❤❡❞✳ ✳ ✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❈✐r❝❧❡ ❍♦♠❡♦♠♦r♣❤✐s♠s ✴ ❝♦♥❥✉❣❛t✐♦♥ ✲ ❛ r❡♠✐♥❞❡r

❋♦r t❤♦s❡ ✇❤♦ ❤❛✈❡ ❢♦r❣♦tt❡♥✿ ✶✮ ■♥ t❤❡ ✶✽✾✵✬s P♦✐♥❝❛ré ♣r♦✈❡❞ t❤❛t ❛♥② ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣ ❝✐r❝❧❡ ❤♦♠❡♦♠♦r♣❤✐s♠ f : S✶ → S✶ ❞❡✜♥❡s ❛ ✉♥✐q✉❡ ♣❛r❛♠❡t❡r α ∈ (✵, ✶] ❝❛❧❧❡❞ t❤❡ r♦t❛t✐♦♥ ♥✉♠❜❡r✱ ❛♥❞ s❤♦✉❧❞ t❤✐s ♥✉♠❜❡r ❜❡ ✐rr❛t✐♦♥❛❧✱ t❤❡♥ t❤❡ ♠❛♣ f ✐s s❡♠✐✲❝♦♥❥✉❣❛t❡ t♦ ❛ r♦t❛t✐♦♥ ❜② α✳ ✷✮ ■♥ t❤❡ ✶✾✸✵✬s ❉❡♥❥♦② ♣r♦✈❡❞ t❤❛t ✐s ✐♥ ❢❛❝t ❵❢✉❧❧②✬ ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✱ ♣r♦✈✐❞❡❞ t❤❛t ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳ ✸✮ ▼♦r❡♦✈❡r✱ ❤❡ ❣❛✈❡ ❡①❛♠♣❧❡s ♦❢

✶ ❞✐✛❡♦♠♦r♣❤✐s♠s t❤❛t ❛r❡ ♥♦t

❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✳ ❍❡r♠❛♥ ✭✐♥ ✶✾✼✾✮ ❣❛✈❡ s✉❝❤ ❡①❛♠♣❧❡s ❢♦r ❛♥②

✶

✱ ✇❤❡r❡ ✵ ✶ ✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❈✐r❝❧❡ ❍♦♠❡♦♠♦r♣❤✐s♠s ✴ ❝♦♥❥✉❣❛t✐♦♥ ✲ ❛ r❡♠✐♥❞❡r

❋♦r t❤♦s❡ ✇❤♦ ❤❛✈❡ ❢♦r❣♦tt❡♥✿ ✶✮ ■♥ t❤❡ ✶✽✾✵✬s P♦✐♥❝❛ré ♣r♦✈❡❞ t❤❛t ❛♥② ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣ ❝✐r❝❧❡ ❤♦♠❡♦♠♦r♣❤✐s♠ f : S✶ → S✶ ❞❡✜♥❡s ❛ ✉♥✐q✉❡ ♣❛r❛♠❡t❡r α ∈ (✵, ✶] ❝❛❧❧❡❞ t❤❡ r♦t❛t✐♦♥ ♥✉♠❜❡r✱ ❛♥❞ s❤♦✉❧❞ t❤✐s ♥✉♠❜❡r ❜❡ ✐rr❛t✐♦♥❛❧✱ t❤❡♥ t❤❡ ♠❛♣ f ✐s s❡♠✐✲❝♦♥❥✉❣❛t❡ t♦ ❛ r♦t❛t✐♦♥ ❜② α✳ ✷✮ ■♥ t❤❡ ✶✾✸✵✬s ❉❡♥❥♦② ♣r♦✈❡❞ t❤❛t f ✐s ✐♥ ❢❛❝t ❵❢✉❧❧②✬ ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✱ ♣r♦✈✐❞❡❞ t❤❛t f ′ ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳ ✸✮ ▼♦r❡♦✈❡r✱ ❤❡ ❣❛✈❡ ❡①❛♠♣❧❡s ♦❢

✶ ❞✐✛❡♦♠♦r♣❤✐s♠s t❤❛t ❛r❡ ♥♦t

❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✳ ❍❡r♠❛♥ ✭✐♥ ✶✾✼✾✮ ❣❛✈❡ s✉❝❤ ❡①❛♠♣❧❡s ❢♦r ❛♥②

✶

✱ ✇❤❡r❡ ✵ ✶ ✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❈✐r❝❧❡ ❍♦♠❡♦♠♦r♣❤✐s♠s ✴ ❝♦♥❥✉❣❛t✐♦♥ ✲ ❛ r❡♠✐♥❞❡r

❋♦r t❤♦s❡ ✇❤♦ ❤❛✈❡ ❢♦r❣♦tt❡♥✿ ✶✮ ■♥ t❤❡ ✶✽✾✵✬s P♦✐♥❝❛ré ♣r♦✈❡❞ t❤❛t ❛♥② ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣ ❝✐r❝❧❡ ❤♦♠❡♦♠♦r♣❤✐s♠ f : S✶ → S✶ ❞❡✜♥❡s ❛ ✉♥✐q✉❡ ♣❛r❛♠❡t❡r α ∈ (✵, ✶] ❝❛❧❧❡❞ t❤❡ r♦t❛t✐♦♥ ♥✉♠❜❡r✱ ❛♥❞ s❤♦✉❧❞ t❤✐s ♥✉♠❜❡r ❜❡ ✐rr❛t✐♦♥❛❧✱ t❤❡♥ t❤❡ ♠❛♣ f ✐s s❡♠✐✲❝♦♥❥✉❣❛t❡ t♦ ❛ r♦t❛t✐♦♥ ❜② α✳ ✷✮ ■♥ t❤❡ ✶✾✸✵✬s ❉❡♥❥♦② ♣r♦✈❡❞ t❤❛t f ✐s ✐♥ ❢❛❝t ❵❢✉❧❧②✬ ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✱ ♣r♦✈✐❞❡❞ t❤❛t f ′ ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳ ✸✮ ▼♦r❡♦✈❡r✱ ❤❡ ❣❛✈❡ ❡①❛♠♣❧❡s ♦❢ C✶ ❞✐✛❡♦♠♦r♣❤✐s♠s t❤❛t ❛r❡ ♥♦t ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✳ ❍❡r♠❛♥ ✭✐♥ ✶✾✼✾✮ ❣❛✈❡ s✉❝❤ ❡①❛♠♣❧❡s ❢♦r ❛♥② C✶+δ✱ ✇❤❡r❡ δ ∈ (✵, ✶)✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❉❡♥❥♦② ♠❛♣

❉❡✜♥✐t✐♦♥ ❲❡ ✇✐❧❧ ❝❛❧❧ ❛♥② ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣ ❝✐r❝❧❡ ❤♦♠❡♦♠♦r♣❤✐s♠ ✇✐t❤ ❛♥ ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ♥✉♠❜❡r t❤❛t ✐s ♥♦t ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥ ❛ ❉❡♥❥♦② ♠❛♣✳ ❉❡✜♥✐t✐♦♥ ❆ s❡t Ω ✐s ❝❛❧❧❡❞ ♠✐♥✐♠❛❧ ✭❢♦r t❤❡ ❤♦♠❡♦♠♦r♣❤✐s♠ f ✮ ✐❢ ✐t ✐s ♥♦♥✲❡♠♣t②✱ ❝♦♠♣❛❝t✱ ✐♥✈❛r✐❛♥t ❛♥❞ ❤❛s ♥♦ ♣r♦♣❡r s✉❜s❡t ✇✐t❤ t❤❡s❡ ♣r♦♣❡rt✐❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ Ω ✐s ♥♦♥✲❡♠♣t②✱ f (Ω) = Ω ❛♥❞ ❡❛❝❤ ❢♦r✇❛r❞ ♦r❜✐t ♦❢ ❛ ♣♦✐♥t x ∈ Ω ✐s ❞❡♥s❡ ✐♥ Ω✳ P♦✐♥❝❛ré✬s r❡s✉❧t ❣✐✈❡s t❤❛t ✐❢ S✶ ✐s ♠✐♥✐♠❛❧✱ t❤❡♥ f ✐s ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❊♥t❡r t❤❡ s❡q✉❡♥❝❡

❖✉r ✐♥t❡r✈❛❧s Jn ✇✐❧❧ ❤❛✈❡ ❧❡♥❣t❤s s❛t✐s❢②✐♥❣✿ ✐✮

• n∈Z

ℓn ≤ ✶✱ ❜✉t ✐♥ ❢❛❝t ✇❡ ✇✐❧❧ ❛ss✉♠❡ ❡q✉❛❧ t♦ ✶✳ ✐✐✮ ❧✐♠

n→±∞

❧♥ |ℓn − ℓn+✶| ❧♥ ℓn = ✶ + δ, ❢♦r s♦♠❡ δ ∈ (✵, ✶)✳ ▼♦❞❡❧ s❡q✉❡♥❝❡ ❆ ♠♦❞❡❧ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ s❡q✉❡♥❝❡ ✐s ℓn = cδ(|n| + ✶)−✶/δ, ✇❤❡r❡ c−✶

δ

=

• n∈Z

(|n| + ✶)−✶/δ

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❚❤❡ ♠❡tr✐❝

❙❡t Ωδ

α = S✶ \

• n∈Z

Jn✳ ▲❡t h: Ωδ

α → S✶ ❜❡ t❤❡ s❡♠✐✲❝♦♥❥✉❣❛❝② ❛♥❞ r❡♠❡♠❜❡r t❤❛t ✇❡

❛ss✉♠❡ h(J✵) = ✵✳ ❋♦r x, y ♥♦t ✐♥ t❤❡ ♦r❜✐t ✭❜② r♦t❛t✐♦♥✮ ♦❢ ✵ h−✶ ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ ✇❡ ❤❛✈❡✿ d(h−✶(x), h−✶(y)) =

• n:nα∈(x,y)

|Jn| =

• n:nα∈(x,y)

ℓn. ❆♥❞ ✐❢ x ✭❛♥❞✴♦r y✮ ❛r❡ ✐♥ t❤❡ ♦r❜✐t ♦❢ ③❡r♦✱ t❤❡♥ h−✶(x) ❝♦♥s✐sts ♦❢ t✇♦ ♣♦✐♥ts ❛♥❞ ✇❡ ❤❛✈❡ t♦ t❛❦❡ t❤❡ ❝♦rr❡❝t ♣r❡✐♠❛❣❡ ✭✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❛r❝ ❜❡t✇❡❡♥ h−✶(x) ❛♥❞ h−✶(y) ✐s t❤❡ s❤♦rt❡st ♣♦ss✐❜❧❡✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❉✐♦♣❤❛♥t✐♥❡ ❝❧❛ss

❉❡✜♥✐t✐♦♥ ❆♥ ✐rr❛t✐♦♥❛❧ α ❤❛s ❛ ❉✐♦♣❤❛♥t✐♥❡ ❝❧❛ss ν > ✵✱ ✐❢ ✐♥❢

p∈Z |qα − p| ≤ ✶

qµ ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② s♦❧✉t✐♦♥ ❢♦r µ < ν ❛♥❞ ❛t ♠♦st ✜♥✐t❡❧② ♠❛♥② ❢♦r µ > ν✳ ❘❡♠❛r❦ ❚❤❡ ❣♦❧❞❡♥ r❛t✐♦ φ ❤❛s ❝❧❛ss ✶✳ ✭❆♥❞ t❤❡ ❝❧❛ss ❝❛♥♥♦t ❜❡ ❧♦✇❡r✮✳ ❚❤❡ ▲✐♦✉✈✐❧❧❡ ♥✉♠❜❡rs ❤❛✈❡ ❝❧❛ss +∞✳ ❚❤❡ s❡t ♦❢ ♣♦✐♥ts ♦❢ ❝❧❛ss ν ❤❛s ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥ ✶

ν ✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

❉❡♥♦t❡ ❜② [a✶, a✷, . . .] t❤❡ st❛♥❞❛r❞ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ α❀ ❛♥❞ ❜② qn t❤❡ ❞❡♥♦♠✐♥❛t♦rs ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥ts ✭✜♥✐t❡ ❢r❛❝t✐♦♥s✮✳ ❘❡❝❛❧❧ t❤❛t qn+✶ = anqn + qn−✶ ❛♥❞ ✶ qn(an + ✷) ≤ ✐♥❢

p∈Z |qnα − p| ≤

✶ anqn . ❙♦ ✇❡ ♠❛② t❤✐♥❦ ♦❢ ❛ ♥✉♠❜❡r ✇✐t❤ ❉✐♦♣❤❛♥t✐♥❡ ❝❧❛ss ν ❛s ♦♥❡ s❛t✐s❢②✐♥❣ ❛ s❡q✉❡♥❝❡ qn+✶ ≈ qν

n.

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❚❤❡ ♦♥❧② t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭❑r❛✕❙❝❤♠❡❧❧✐♥❣✮ ❆ss✉♠❡ δ ∈ (✵, ✶) ❛♥❞ α ❤❛s ❉✐♦♣❤❛♥t✐♥❡ ❝❧❛ss ν✳ ❚❤❡♥ ❛♥ ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣ C✶+δ ❞✐✛❡♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❝✐r❝❧❡ ✇✐t❤ r♦t❛t✐♦♥ ♥✉♠❜❡r α ❛♥❞ t❤❡ ♠✐♥✐♠❛❧ s❡t Ωδ

α s❛t✐s✜❡s

❞✐♠B Ωδ

α ≥ δ

❛♥❞ ❞✐♠H Ωδ

α ≥ δ

ν , ❛♥❞ t❛❦✐♥❣ t❤❡ ♠♦❞❡❧ s❡q✉❡♥❝❡ ❣✐✈❡s ❡q✉❛❧✐t✐❡s✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ✉♣♣❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡

❲❡ ✇❛♥t t♦ ♣r♦✈❡ ❧✐♠ s✉♣

r→+∞

❧♦❣ N(r) − ❧♦❣(r) ≤ δ ❢♦r ❛ r❡❛s♦♥❛❜❧② ❞❡♥s❡ s❡q✉❡♥❝❡ ♦❢ r✬s✳ ❋✐① ❛♥❞ ❝♦♥s✐❞❡r

✶

✳ ❚❤✐s ✐s ❛ s✉♠ ♦❢ ✷ ✶ ❞✐s❥♦✐♥t ✐♥t❡r✈❛❧s ♦❢ t♦t❛❧ ❧❡♥❣t❤ ✶

✶ ✶

✳ ❚❤✐s s❡t ♠❛② ❜❡ ❝♦✈❡r❡❞ ❛❧♠♦st tr✐✈✐❛❧❧② ❜② ✐♥t❡r✈❛❧s ♦❢ ❛✈❡r❛❣❡ ❧❡♥❣t❤✱ ✐✳❡✳

✶ ✶

✷ ✶ ✳ ❍♦✇ ♠❛♥② s✉❝❤ ✐♥t❡r✈❛❧s ❞♦ ✇❡ ♥❡❡❞❄

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ✉♣♣❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡

❲❡ ✇❛♥t t♦ ♣r♦✈❡ ❧✐♠ s✉♣

r→+∞

❧♦❣ N(r) − ❧♦❣(r) ≤ δ ❢♦r ❛ r❡❛s♦♥❛❜❧② ❞❡♥s❡ s❡q✉❡♥❝❡ ♦❢ r✬s✳ ❋✐① n ∈ N ❛♥❞ ❝♦♥s✐❞❡r Jn = S✶ \

• −n≤k≤n

Jk

• ✳ ❚❤✐s ✐s ❛ s✉♠ ♦❢

✷n + ✶ ❞✐s❥♦✐♥t ✐♥t❡r✈❛❧s ♦❢ t♦t❛❧ ❧❡♥❣t❤

• ✶ −
• −n≤k≤n

ℓk

• ≈ n✶−✶/δ✳

❚❤✐s s❡t ♠❛② ❜❡ ❝♦✈❡r❡❞ ❛❧♠♦st tr✐✈✐❛❧❧② ❜② ✐♥t❡r✈❛❧s ♦❢ ❛✈❡r❛❣❡ ❧❡♥❣t❤✱ ✐✳❡✳

✶ ✶

✷ ✶ ✳ ❍♦✇ ♠❛♥② s✉❝❤ ✐♥t❡r✈❛❧s ❞♦ ✇❡ ♥❡❡❞❄

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ✉♣♣❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡

❲❡ ✇❛♥t t♦ ♣r♦✈❡ ❧✐♠ s✉♣

r→+∞

❧♦❣ N(r) − ❧♦❣(r) ≤ δ ❢♦r ❛ r❡❛s♦♥❛❜❧② ❞❡♥s❡ s❡q✉❡♥❝❡ ♦❢ r✬s✳ ❋✐① n ∈ N ❛♥❞ ❝♦♥s✐❞❡r Jn = S✶ \

• −n≤k≤n

Jk

• ✳ ❚❤✐s ✐s ❛ s✉♠ ♦❢

✷n + ✶ ❞✐s❥♦✐♥t ✐♥t❡r✈❛❧s ♦❢ t♦t❛❧ ❧❡♥❣t❤

• ✶ −
• −n≤k≤n

ℓk

• ≈ n✶−✶/δ✳

❚❤✐s s❡t ♠❛② ❜❡ ❝♦✈❡r❡❞ ❛❧♠♦st tr✐✈✐❛❧❧② ❜② ✐♥t❡r✈❛❧s ♦❢ ❛✈❡r❛❣❡ ❧❡♥❣t❤✱ ✐✳❡✳ n✶−✶/δ/(✷n + ✶)✳ ❍♦✇ ♠❛♥② s✉❝❤ ✐♥t❡r✈❛❧s ❞♦ ✇❡ ♥❡❡❞❄

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ✉♣♣❡r ❜♦✉♥❞✱ ❝♦♥t✳

❲❡ ♥❡❡❞ ✶ t♦ ❝♦✈❡r ❡✈❡r② s❡t ♦❢ ❧❡♥❣t❤ s❤♦rt❡r t❤❛♥ t❤❡ ❛✈❡r❛❣❡ ✭❛t ♠♦st ✷n + ✶ s❡ts✮✳ ❆♥❞ ❛t ♠♦st t✇✐❝❡ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ❢♦r t❤❡ ❧♦♥❣❡r s❡ts✳ ❚❤✐s ②✐❡❧❞s N(n✶−✶/δ/(✷n + ✶)) ≤ ✻n + ✸. P❧✉❣❣✐♥❣ t❤✐s ✐♥t♦ t❤❡ ❞✐♠❡♥s✐♦♥ ❢♦r♠✉❧❛ ❣✐✈❡s ❧♦❣ ❧♦❣ ❧♦❣ ✻ ✸ ❧♦❣

✶ ✶

✷ ✶

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ✉♣♣❡r ❜♦✉♥❞✱ ❝♦♥t✳

❲❡ ♥❡❡❞ ✶ t♦ ❝♦✈❡r ❡✈❡r② s❡t ♦❢ ❧❡♥❣t❤ s❤♦rt❡r t❤❛♥ t❤❡ ❛✈❡r❛❣❡ ✭❛t ♠♦st ✷n + ✶ s❡ts✮✳ ❆♥❞ ❛t ♠♦st t✇✐❝❡ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ❢♦r t❤❡ ❧♦♥❣❡r s❡ts✳ ❚❤✐s ②✐❡❧❞s N(n✶−✶/δ/(✷n + ✶)) ≤ ✻n + ✸. P❧✉❣❣✐♥❣ t❤✐s ✐♥t♦ t❤❡ ❞✐♠❡♥s✐♦♥ ❢♦r♠✉❧❛ ❣✐✈❡s ❧♦❣ N(r) − ❧♦❣(r) ≤ ❧♦❣(✻n + ✸) − ❧♦❣

• n✶−✶/δ/(✷n + ✶)

≈ δ

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ❧♦✇❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡

❋✐① m, n ∈ N ❛♥❞ ♦❜s❡r✈❡ t❤❡ s❡t Z = S✶ \

• −m≤k≤n

Jk

• ✳ ❚❤❡ s❡t

{lα : −m − n − ✶ ≤ l ≤ m + n + ✶} ❝♦♥t❛✐♥s ❛t ❧❡❛st ♦♥❡ ♣♦✐♥t ✐♥ ❡❛❝❤ ♦❢ t❤❡ m + n + ✶ ✐♥t❡r✈❛❧s ♦❢ Z✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ ❛♥② ♦❢ t❤♦s❡ ✐♥t❡r✈❛❧s ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❜② cδ(✷n + ✷m + ✸)−✶/δ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ♦❢ ❛♥② t✇♦ ✐♥t❡r✈❛❧s ✐s ❛t ❧❡❛st cδ(♠❛①(n, m))−✶/δ✳ ❙♦ t♦ ❝♦✈❡r ❜② ✐♥t❡r✈❛❧s ♦❢ ❧❡♥❣t❤ ✷ ✷ ✸

✶

✇❡ ♠✉st ✉s❡ ❛t ❧❡❛st ✶ ✐♥t❡r✈❛❧s✳ ❚❤✐s ♣r♦✈❡s t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡✳ ❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✇❡ ♥❡❡❞ t♦ ❛❞❞✐t✐♦♥❛❧❧② ♣r♦✈❡ t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥ ❣✐✈❡ t❤❛t

✶

❢♦r ❛❧❧ ✵ ✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❇♦① ❞✐♠❡♥s✐♦♥ ✲ ❧♦✇❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡

❋✐① m, n ∈ N ❛♥❞ ♦❜s❡r✈❡ t❤❡ s❡t Z = S✶ \

• −m≤k≤n

Jk

• ✳ ❚❤❡ s❡t

{lα : −m − n − ✶ ≤ l ≤ m + n + ✶} ❝♦♥t❛✐♥s ❛t ❧❡❛st ♦♥❡ ♣♦✐♥t ✐♥ ❡❛❝❤ ♦❢ t❤❡ m + n + ✶ ✐♥t❡r✈❛❧s ♦❢ Z✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ ❛♥② ♦❢ t❤♦s❡ ✐♥t❡r✈❛❧s ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❜② cδ(✷n + ✷m + ✸)−✶/δ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ♦❢ ❛♥② t✇♦ ✐♥t❡r✈❛❧s ✐s ❛t ❧❡❛st cδ(♠❛①(n, m))−✶/δ✳ ❙♦ t♦ ❝♦✈❡r Ω ❜② ✐♥t❡r✈❛❧s ♦❢ ❧❡♥❣t❤ cδ(✷n + ✷m + ✸)−✶/δ ✇❡ ♠✉st ✉s❡ ❛t ❧❡❛st m + n + ✶ ✐♥t❡r✈❛❧s✳ ❚❤✐s ♣r♦✈❡s t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦r ♠♦❞❡❧ ❝❛s❡✳ ❋♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✇❡ ♥❡❡❞ t♦ ❛❞❞✐t✐♦♥❛❧❧② ♣r♦✈❡ t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥ ℓn ❣✐✈❡ t❤❛t ℓn > n−✶/θ ❢♦r ❛❧❧ ✵ < θ < δ✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❉✐♠❡♥s✐♦♥ ❜② r❡❝✉rr❡♥❝❡

❍❡r❡ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t (X, d) ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ T : X → X ❛ ❇♦r❡❧ ♠❡❛s✉r❛❜❧❡ ♠❛♣❀ µ ✐s ❛ T✕✐♥✈❛r✐❛♥t✱ ❡r❣♦❞✐❝✱ ♣r♦❜❛❜✐❧✐t②✱ ❇♦r❡❧ ♠❡❛s✉r❡ ♦♥ X✳ ❚❤❡♦r❡♠ ❲✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❞②♥❛♠✐❝❛❧ s②st❡♠ ❛s ❛❜♦✈❡✱ ❢♦r ❛♥② β > ✵ ❛♥❞ ❢♦r µ ✕ ❛❧♠♦st ❡✈❡r② x ∈ X ✇❡ ❤❛✈❡ ❧✐♠ ✐♥❢

n→∞ n✶/βd(T n(x), x) ≤ g(x)✶/β, ✇❤❡r❡ g(x) = ❧✐♠ s✉♣ r→✵ Hβ(Bx(r)) µ(Bx(r)) ✳

❘❡♠❛r❦ ◆♦t❡ t❤❛t g(x) ♠❛② ❜❡ ❡q✉❛❧ t♦ ✵ ♦r +∞✳ ❚❤❡ st❛t❡♠❡♥t st✐❧❧ ❤♦❧❞s✳

❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts