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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

✷✮ ■♥ t❤❡ ✶✾✸✵✬s ❉❡♥❥♦② ♣r♦✈❡❞ t❤❛t ✐s ✐♥ ❢❛❝t ❵❢✉❧❧②✬ ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✱ ♣r♦✈✐❞❡❞ t❤❛t ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳ ✶ ❞✐✛❡♦♠♦r♣❤✐s♠s t❤❛t ❛r❡ ♥♦t ✸✮ ▼♦r❡♦✈❡r✱ ❤❡ ❣❛✈❡ ❡①❛♠♣❧❡s ♦❢ ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✳ ❍❡r♠❛♥ ✭✐♥ ✶✾✼✾✮ ❣❛✈❡ s✉❝❤ ✶ ❡①❛♠♣❧❡s ❢♦r ❛♥② ✱ ✇❤❡r❡ ✵ ✶ ✳ ❈✐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 ✳ ❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

✶ ❞✐✛❡♦♠♦r♣❤✐s♠s t❤❛t ❛r❡ ♥♦t ✸✮ ▼♦r❡♦✈❡r✱ ❤❡ ❣❛✈❡ ❡①❛♠♣❧❡s ♦❢ ❝♦♥❥✉❣❛t❡ t♦ t❤❡ r♦t❛t✐♦♥✳ ❍❡r♠❛♥ ✭✐♥ ✶✾✼✾✮ ❣❛✈❡ s✉❝❤ ✶ ❡①❛♠♣❧❡s ❢♦r ❛♥② ✱ ✇❤❡r❡ ✵ ✶ ✳ ❈✐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✐♦♥✳ 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 J n ✇✐❧❧ ❤❛✈❡ ❧❡♥❣t❤s s❛t✐s❢②✐♥❣✿ � ✐✮ ℓ n ≤ ✶✱ ❜✉t ✐♥ ❢❛❝t ✇❡ ✇✐❧❧ ❛ss✉♠❡ ❡q✉❛❧ t♦ ✶✳ n ∈ Z ❧♥ | ℓ n − ℓ n + ✶ | ✐✐✮ ❧✐♠ = ✶ + δ, ❧♥ ℓ n n →±∞ ❢♦r s♦♠❡ δ ∈ ( ✵ , ✶ ) ✳ ▼♦❞❡❧ s❡q✉❡♥❝❡ ❆ ♠♦❞❡❧ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ s❡q✉❡♥❝❡ ✐s � ℓ n = c δ ( | n | + ✶ ) − ✶ /δ , ✇❤❡r❡ c − ✶ ( | n | + ✶ ) − ✶ /δ = δ n ∈ Z ❾✳ P❛✇❡❧❡❝ ❉❡♥❥♦② s❡ts

❚❤❡ ♠❡tr✐❝ α = S ✶ \ ❙❡t Ω δ � J n ✳ n ∈ Z α → S ✶ ❜❡ t❤❡ s❡♠✐✲❝♦♥❥✉❣❛❝② ❛♥❞ r❡♠❡♠❜❡r t❤❛t ✇❡ ▲❡t h : Ω δ ❛ss✉♠❡ h ( J ✵ ) = ✵✳ ❋♦r x , y ♥♦t ✐♥ t❤❡ ♦r❜✐t ✭❜② r♦t❛t✐♦♥✮ ♦❢ ✵ h − ✶ ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ ✇❡ ❤❛✈❡✿ � � d ( h − ✶ ( x ) , h − ✶ ( y )) = | J n | = ℓ n . n : n α ∈ ( x , y ) n : n α ∈ ( x , y ) ❆♥❞ ✐❢ 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 | q α − p | ≤ ✶ ✐♥❢ q µ ❤❛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