t s r rtts stst - - PowerPoint PPT Presentation

❙t❡♣ ❝♦❝②❝❧❡s ♦✈❡r r♦t❛t✐♦♥s ✿ st♦❝❤❛st✐❝ ♣r♦♣❡rt✐❡s ❏❡❛♥✲P✐❡rr❡ ❈♦♥③❡ ✭❯♥✐✈❡rs✐t② ♦❢ ❘❡♥♥❡s ✶✮ ❆❜str❛❝t ✿ ▲❡t x → x + α ❜❡ ❛ r♦t❛t✐♦♥ ♦♥ t❤❡ ❝✐r❝❧❡ ❛♥❞ ❧❡t ϕ ❜❡ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳ ❉❡♥♦t❡ ❜② ϕn(x) := n−1

j=0 ϕ(x+jα)

t❤❡ ❡r❣♦❞✐❝ s✉♠s✳ ❲❤❡♥ α ❤❛s ❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥ts✱ ❛♥ ❛✳s✳ ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❤♦❧❞s ❛❧♦♥❣ ❛ s✉❜s❡q✉❡♥❝❡✳ ❲❤❡♥ α ✐s ♦❢ ❜♦✉♥❞❡❞ t②♣❡✱ ❛♥❞ ✉♥❞❡r ❛ ❞✐♦♣❤❛♥t✐♥❡ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s✱ ❢♦r ❛ s❡t ♦❢ t✐♠❡s n ♦❢ ❞❡♥s✐t② ✶✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ϕn ❛❢t❡r ♥♦r♠❛❧✐③❛t✐♦♥ ✐s ❝❧♦s❡ t♦ t❤❡ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥✳ ❚❤✐s ❢♦❧❧♦✇s ❢r♦♠ ❞❡❝♦rr❡❧❛t✐♦♥ ✐♥❡q✉❛❧✐t✐❡s ❜❡t✇❡❡♥ t❤❡ ❡r❣♦❞✐❝ s✉♠s ❛t t✐♠❡ qk✱ ✇❤❡r❡ t❤❡ qk✬s ❛r❡ t❤❡ ❞❡♥♦♠✐♥❛t♦rs ♦❢ α✱ ✇❤✐❝❤ ✐s ✈❛❧✐❞ ❢♦r ❛ ❝❧❛ss ♦❢ α✬s ✐♥❝❧✉❞✐♥❣ ✐rr❛t✐♦♥❛❧s ✇✐t❤ ❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥ts✳ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❙té♣❤❛♥❡ ▲❡ ❇♦r❣♥❡✮ ❋♦r ▼❛r✐✉s③ ▲❡♠❛➠❝③②❦ ✻✵t❤ ❜✐rt❤❞❛② ❇❡❞❧❡✇♦ ✶✹ ❏✉♥❡ ✷✵✶✽

✶

■♥tr♦❞✉❝t✐♦♥ ▼❛♥② r❡s✉❧ts ❧✐♥❦ t❤❡ st♦❝❤❛st✐❝✐t② ♦❢ ❛ ❞②♥❛♠✐❝❛❧ s②st❡♠ t♦ ❧✐♠✐t t❤❡♦r❡♠s ✐♥ ❞✐str✐❜✉t✐♦♥ ❢♦r t❤❡ ❡r❣♦❞✐❝ s✉♠s ♦❢ ❛♥ ♦❜s❡r✈❛❜❧❡ ϕ✳ ❚❤❡ s✐♠♣❧❡st ❡①❛♠♣❧❡ ✐s T : x → 2x mod 1 ♦♥ X = R/Z ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ✿ t❤❡ ♥♦r♠❛❧✐③❡❞ ❡r❣♦❞✐❝ s✉♠s s❛t✐s❢② ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ✭❈▲❚✮ ✇❤❡♥ ϕ ✐s ❍ö❧❞❡r ♦r ✇✐t❤ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳ ❲❤❡♥ T ✐s ❛♥ ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ x → x + α mod 1 ♦♥ X✱ t❤❡ ❜❡✲ ❤❛✈✐♦✉r ✐s q✉✐t❡ ❞✐✛❡r❡♥t✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❉✐♦♣❤❛♥t✐♥❡ ♣r♦♣❡rt✐❡s ♦❢ α✱ t♦♦ ♠✉❝❤ r❡❣✉❧❛r✐t② ❢♦r ϕ ❝❛♥ ✐♠♣❧② t❤❛t ϕ ✐s ❛ ❝♦❜♦✉♥❞❛r②✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♥❛t✉r❛❧ t♦ ❝♦♥s✐❞❡r ❧❡ss r❡❣✉❧❛r ❜✉t ❇❱ ✭❜♦✉♥✲ ❞❡❞ ✈❛r✐❛t✐♦♥✮ ❢✉♥❝t✐♦♥s✱ ✐♥ ♣❛rt✐❝✉❧❛r st❡♣ ❢✉♥❝t✐♦♥s✳ ◆❡✈❡rt❤❡✲ ❧❡ss✱ ❜② t❤❡ ❉❡♥❥♦②✲❑♦❦s♠❛ ✐♥❡q✉❛❧✐t②✱ t❤❡ ❡r❣♦❞✐❝ s✉♠s ϕL(x) =

L−1

ϕ(x+jα) ♦❢ ❛ ❇❱ ❢✉♥❝t✐♦♥ ϕ ❛r❡ ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ❛❧♦♥❣ t❤❡ s❡q✉❡♥❝❡ (qn) ♦❢ ❞❡♥♦♠✐♥❛t♦rs ♦❢ α✳ ❇✉t ♦♥❡ ❝❛♥ ❛s❦ ✐❢ ❛❧♦♥❣ ♦t❤❡r s❡q✉❡♥❝❡s ♦❢ t✐♠❡ (Ln) t❤❡r❡ ✐s ❛ ❞✐✛✉s✐✈❡ ❜❡❤❛✈✐♦✉r ❛t s♦♠❡ s❝❛❧❡ ❢♦r t❤❡ ❡r❣♦❞✐❝ s✉♠s✳

✷

❘❡s✉❧ts ♦♥ t❤❡ ❈▲❚ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❋♦✉r✐❡r s❡r✐❡s✱ ✇❤✐❝❤ ❛r❡ r❡✲ ❧❛t❡❞ t♦ ♦✉r ❢r❛♠❡✇♦r❦✱ tr❛❝❡ ❜❛❝❦ t♦ ❙❛❧❡♠ ❛♥❞ ❩②❣♠✉♥❞ ✭✶✾✹✽✮ ✐♥ t❤❡ ✹✵✬s✳ ▼✳ ❉❡♥❦❡r ❛♥❞ ❘✳ ❇✉rt♦♥ ✐♥ ✶✾✽✼✱ t❤❡♥ ▼✳ ❲❡❜❡r✱ ▼✳ ▲❛❝❡② ❛♥❞ ♦t❤❡r ❛✉t❤♦rs ❣❛✈❡ r❡s✉❧ts ♦♥ ❛ ❈▲❚ ❢♦r ❡r❣♦❞✐❝ s✉♠s ❣❡♥❡r❛t❡❞ ❜② r♦t❛t✐♦♥s✳ ❚❤❡✐r ❣♦❛❧ ✇❛s t❤❡ ❡①✐st❡♥❝❡ ❢✉♥❝✲ t✐♦♥s✱ ♥❡❝❡ss❛r✐❧② ✐rr❡❣✉❧❛r✱ ✇❤♦s❡ ❡r❣♦❞✐❝ s✉♠s s❛t✐s❢② ❛ ❈▲❚ ❛❢t❡r s❡❧❢✲♥♦r♠❛❧✐③❛t✐♦♥✳ ■♥ ✶✾✾✼ ❉✳ ❱♦❧♥ý ❛♥❞ P✳ ▲✐❛r❞❡t s❤♦✇❡❞ t❤❛t ❢♦r ❛♥ ❛♣❡r✐♦❞✐❝ ♠❡❛✲ s✉r❡ ♣r❡s❡r✈✐♥❣ s②st❡♠ (X, µ, T) ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ X ❛♥❞ ❢♦r ❛ Gδ s❡t ♦❢ f ✐♥ C0(X) t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s c−1

n

n−1

j=0 f ◦ T j✱ cn ↑ ∞ ❛♥❞ cn/n → 0✱ ❛r❡ ❞❡♥s❡ ✐♥ t❤❡ s❡t ♦❢ ❛❧❧

♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ♦♥ t❤❡ r❡❛❧ ❧✐♥❡✳ ❚❤❡ ❧✐♠✐t t❤❡♦r❡♠s ❛❧♦♥❣ s♦♠❡ s✉❜s❡q✉❡♥❝❡s t❤❛t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ s❤♦✇ ❛r❡ ❢♦r s✐♠♣❧❡ st❡♣s ❢✉♥❝t✐♦♥s✳ ■♥ t❤✐s ❞✐r❡❝t✐♦♥✱ ❢♦r ψ := 1[0,1

2[ − 1[1 2,0[✱ ❋✳ ❍✉✈❡♥❡❡rs ✐♥ ✷✵✵✾ ♣r♦✈❡❞ t❤❛t ❢♦r ❡✈❡r② ✐rr❛t✐♦♥❛❧

α t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Ln)n∈N s✉❝❤ t❤❛t ψLn/√n ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞✳

✸

❍❡r❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❞✐✛✉s✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❡r❣♦❞✐❝ s✉♠s ♦✈❡r ❛ r♦t❛t✐♦♥ ❜② α ♦❢ ❇❱ ❢✉♥❝t✐♦♥s✱ ❧✐❦❡ st❡♣ ❢✉♥❝t✐♦♥s ϕ ✇✐t❤ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❞✐s❝♦♥t✐♥✉✐t✐❡s✳ ❚✇♦ ❞✐✛❡r❡♥t ♠❡t❤♦❞s ❝❛♥ ❜❡ ✉s❡❞ ✿ ✲ ❆ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❧❛❝✉♥❛r② s❡r✐❡s ❧❡❛❞✐♥❣ t♦ ❛♥ ❆❙■P ✭❛❧♠♦st s✉r❡ ✐♥✈❛r✐❛♥❝❡ ♣r✐♥❝✐♣❧❡✮ ❢♦r s✉❜s❡q✉❡♥❝❡s ♦❢ ❡r❣♦❞✐❝ s✉♠s ❢♦r ❇❱ ♦❜s❡r✈❛❜❧❡s ✇❤❡♥ α ❤❛s ✉♥❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥ts✳ ■t ✐s ❜❛s❡❞ ♦♥ ❛ ❧✐♥❦ ❜❡t✇❡❡♥ r♦t❛t✐♦♥s ❛♥❞ ❡①♣❛♥s✐✈❡ ♠❛♣s ✇❤✐❝❤ ❛❧✲ ❧♦✇s t♦ ✉s❡ t❤❡ st♦❝❤❛st✐❝ ❜❡❤❛✈✐♦✉r ♦❢ s✉♠s ♦❢ t❤❡ ❢♦r♠ n

1 fj(kjx)

✇❤❡r❡ (kj) ✐s ❛ ❢❛st ❣r♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs ❛♥❞ (fj) ❛ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✐♥ ❛ ❝❧❛ss ✇❤✐❝❤ ❝♦♥t❛✐♥s t❤❡ ❇❱ ❢✉♥❝t✐♦♥s✳ ■t ✉s❡s ❛ r❡s✉❧t ♦❢ ❇❡r❦❡s ❛♥❞ P❤✐❧✐♣♣ ✭✶✾✼✾✮ ✐♥ ❛ s❧✐❣❤t❧② ❡①t❡♥❞❡❞ ✈❡rs✐♦♥✳ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❙✳ ■s♦❧❛ ❛♥❞ ❙✳ ▲❡ ❇♦r❣♥❡✮✳ ✲ ❆ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ ❞❡❝♦rr❡❧❛t✐♦♥ ✐♥❡q✉❛❧✐t✐❡s ❛♥❞ ✇❡❧❧ ❛❞❛♣t❡❞ t♦ t❤❡ ❜♦✉♥❞❡❞ t②♣❡ ❝❛s❡✳ ■t r❡❧✐❡s ♦♥ ❛♥ ❛❜str❛❝t ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ✈❛❧✐❞ ✉♥❞❡r s♦♠❡ s✉✐t❛❜❧❡ ❞❡❝♦rr❡❧❛t✐♦♥ ❝♦♥❞✐t✐♦♥s✳ ❇❡s✐❞❡ t❤❡ r❡♠❛r❦❛❜❧❡ r❡❝❡♥t ✏t❡♠♣♦r❛❧✑ ❧✐♠✐t t❤❡♦r❡♠s ❢♦r r♦t❛✲ t✐♦♥s ✭❏✳ ❇❡❝❦✱ ▼✳ ❇r♦♠❜❡r❣✱ ❈✳ ❯❧❝✐❣r❛✐✱ ❉✳ ❉♦❧❣♣❛②t✱ ❖✳ ❙❛r✐❣✮✱ t❤❡ s❡❝♦♥❞ ♠❡t❤♦❞ s❤♦✇s t❤❛t ❛ ✏s♣❛t✐❛❧✑ ❛s②♠♣t♦t✐❝ ♥♦r♠❛❧ ❞✐s✲ tr✐❜✉t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ ♦❜s❡r✈❡❞✱ ❢♦r t✐♠❡s r❡str✐❝t❡❞ t♦ ❛ ❧❛r❣❡ s❡t ♦❢ ✐♥t❡❣❡rs✳

✹

✷✳ Pr❡❧✐♠✐♥❛r✐❡s ◆♦t❛t✐♦♥s✱ ❢r❛♠❡✇♦r❦ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱ α ✇✐❧❧ ❜❡ ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ✐♥ ]0, 1[✳ ■ts ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ✐s ❞❡♥♦t❡❞ ❜② α = [0; a1, a2, ..., an, ...]✳ ❋♦r u ∈ R✱ s❡t u := infn∈Z |u − n|✳ ▲❡t (pn/qn)n≥0 ❜❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥ts ♦❢ α✳ ❚❤❡ ✐♥t❡❣❡rs pn, qn ❛r❡ t❤❡ ♥✉♠❡r❛t♦rs ❛♥❞ ❞❡♥♦♠✐♥❛t♦rs ♦❢ α✳ α ✐s s❛✐❞ ♦❢ ❝♦♥st❛♥t t②♣❡ ✭♦r ❤❛s ❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥t ✭❜♣q✮✮✱ ✐❢ supk ak < ∞✳ ❚❤❡ ✉♥✐❢♦r♠ ♠❡❛s✉r❡ ♦♥ T1 ✐❞❡♥t✐✜❡❞ ✇✐t❤ X = [0, 1[ ✐s ❞❡♥♦t❡❞ ❜② µ✳ ❚❤❡ ❛r❣✉♠❡♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ t❛❦❡♥ ♠♦❞✉❧♦ ✶✳ ❋♦r ❛ ✶✲ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥ ϕ✱ ✇❡ ❞❡♥♦t❡ ❜② V (ϕ) t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ ϕ ❝♦♠♣✉t❡❞ ❢♦r ✐ts r❡str✐❝t✐♦♥ t♦ t❤❡ ✐♥t❡r✈❛❧ [0, 1[ ❛♥❞ ✉s❡ t❤❡ s❤♦rt❤❛♥❞ ❇❱ ❢♦r ✏❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✑✳

✺

▲❡t C ❜❡ t❤❡ ❝❧❛ss ♦❢ ❝❡♥t❡r❡❞ ❇❱ ❢✉♥❝t✐♦♥s✳ ■❢ ϕ ✐s ✐♥ C✱ ✐ts ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts cr(ϕ) s❛t✐s❢② ✿ cr(ϕ) = γr(ϕ) r , with K(ϕ) := sup

r=0

|γr(ϕ)| < +∞. ✭✶✮ ❚❤❡ ❝❧❛ss C ❝♦♥t❛✐♥s ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ st❡♣ ❢✉♥❝t✐♦♥s ✇✐t❤ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❞✐s❝♦♥t✐♥✉✐t✐❡s✳ ▲❡t ϕ ❜❡ ✐♥ C✳ ❚❤❡ ❡r❣♦❞✐❝ s✉♠s ℓ−1

j=0 ϕ(x + jα) ❛r❡ ❞❡♥♦t❡❞ ❜② ϕℓ(x)✳

❇② ❉❡♥❥♦②✲❑♦❦s♠❛ ✐♥❡q✉❛❧✐t② ✇❡ ❤❛✈❡ ϕqn∞ = sup

x | qn−1

• ℓ=0

ϕ(x + ℓα)| ≤ V (ϕ). ✭✷✮ ❚❤❡r❡❢♦r❡✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❡r❣♦❞✐❝ s✉♠s ϕℓ ❞❡♣❡♥❞s str♦♥❣❧② ♦♥ t❤❡ ✈❛❧✉❡s ♦❢ ℓ✱ s✐♥❝❡ ❢♦r ❛ ❇❱ ❝❡♥t❡r❡❞ ❢✉♥❝t✐♦♥✱ t❤❡ ❡r❣♦❞✐❝ s✉♠s ❛r❡ ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ❛❧♦♥❣ t❤❡ s❡q✉❡♥❝❡ (qn) ♦❢ ❞❡♥♦♠✐♥❛t♦rs ♦❢ α✳ ❚❤✐s ❝♦♥tr❛sts ✇✐t❤ t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❡r❣♦❞✐❝ s✉♠s ♦❢ ❛ ❝❡♥t❡r❡❞ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ ♦❢ ❛ ❤②♣❡r❜♦❧✐❝ ♠❛♣✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ r♦t❛t✐♦♥✱ t❤❡r❡ ✐s ❛ ❝♦♠♣❛❝t ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ✵ s✉❝❤ t❤❛t r❡❝✉rr❡♥❝❡ ♦❢ t❤❡ ❡r❣♦❞✐❝ s✉♠s ♦❝❝✉rs ❛❧♦♥❣ ❛ s❡q✉❡♥❝❡ ♦❢ t✐♠❡s ♥♦t ❞❡♣❡♥❞✐♥❣ ♦♥ x✳

✻

❖str♦✇s❦✐ ❡①♣❛♥s✐♦♥ ▲❡t ✉s r❡❝❛❧❧ t❤❡ ❖str♦✇s❦✐ ❡①♣❛♥s✐♦♥✳ ❋♦r N ≥ 1 ✇❡ ♣✉t m = m(N) := ℓ, ✐❢ N ∈ [qℓ, qℓ+1[. ✭✸✮ ❬❘❡♠❛r❦ t❤❛t ✐❢ α ❤❛s ❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥ts ✭K = sup an < ∞✮✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t C > 0 s✉❝❤ t❤❛t C−1 ≤ ln N/m(N) ≤ C✳❪ ❲❡ ❝❛♥ ✇r✐t❡ N = bmqm + r✱ ✇✐t❤ 1 ≤ bm ≤ am+1✱ 0 ≤ r < qm✳ ❇② ✐t❡r❛t✐♦♥✱ ✇❡ ❣❡t ❢♦r N t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ✿ N =

m

• k=0

bk qk, ✇✐t❤ 0 ≤ bk ≤ ak+1 ❢♦r 1 ≤ k < m, ❛♥❞ 0 ≤ b0 ≤ a1 − 1✱ 1 ≤ bm ≤ am+1✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❡r❣♦❞✐❝ s✉♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✿ ϕN(x) =

m

• ℓ=0

Nℓ−1

• j=Nℓ−1

ϕ(x + jα) =

m

• ℓ=0

ϕbℓ qℓ(x + Nℓ−1α), ✭✹✮ ✇✐t❤ N0 = b0✱ Nℓ = ℓ

k=0 bk qk ❢♦r ℓ ≤ m✳

✼

❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❛❜♦✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ❡r❣♦❞✐❝ s✉♠s ϕqn ❛t t✐♠❡ qn ✇✐❧❧ ♣❧❛② t❤❡ r♦❧❡ ♦❢ ❛t♦♠s ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ s✉♠ ϕN ❛s ❛ s✉♠ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❖♥❡ ♦❢ t❤❡ ♠❡t❤♦❞s ♦❢ ♣r♦♦❢ ♦❢ ❛ ❈▲❚ r❡❧✐❡s ♦♥ ❛ ❞❡❝♦rr❡❧❛t✐♦♥ ♣r♦♣❡rt② ❜❡t✇❡❡♥ ϕqn ❛♥❞ ϕqn′✳ ❚❤❡ ❛✐♠ ■♥ ✈✐❡✇ ♦❢ t❤❡ ❖str♦✇s❦✐ ❡①♣❛♥s✐♦♥✱ ✇❡ ❝❛♥ ❛s❦ ✐❢ t❤❡r❡ ✐s ❛ ❞✐✛✉s✐✈❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❡r❣♦❞✐❝ s✉♠s ❛❧♦♥❣ s✉✐t❛❜❧❡ s✉❜s❡q✉❡♥❝❡s ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ qn✬s✳ ❉✐✛❡r❡♥t ♠❡t❤♦❞s ❝❛♥ ❜❡ ✉s❡❞✳

• ❖♥❡ ♠❡t❤♦❞ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ■❢ an ✐s ❜✐❣✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t

t❤❡r❡ ✐s fn s✉❝❤ t❤❛t t❤❡ 1

qn−♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥ fn(qn.) ❛♣♣r♦①✐♠❛t❡s

✇❡❧❧ t❤❡ ❡r❣♦❞✐❝ s✉♠ ϕqn✳ ❲❤❡♥ (an)✱ ♦r ❛ s✉❜s❡q✉❡♥❝❡ ♦❢ (an)✱ ✐s ❣r♦✇✐♥❣ ❢❛st✱ ❛ ❈▲❚ ❢♦r s✉❜s❡q✉❡♥❝❡s (ϕLn) ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ st♦❝❤❛st✐❝ ♣r♦♣❡rt✐❡s ♦❢ s✉♠s ♦❢ t❤❡ ❢♦r♠ N

k=1 fnk(qnk.)✳ ❲❡

✉s❡ t❤❡♥ ❛ r❡s✉❧t ♦❢ ❇❡r❦❡s ❛♥❞ P❤✐❧✐♣♣ ✇❤✐❝❤ ♣r♦✈✐❞❡s ❛ ❈▲❚ ❛♥❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❜② ❛ ❲✐❡♥❡r ♣r♦❝❡ss ❢♦r s✉♠s ♦❢ t❤✐s ❢♦r♠✳

✽

• ❆♥♦t❤❡r ♠❡t❤♦❞ ✐s t♦ s❤♦✇ ❞❡❝♦rr❡❧❛t✐♦♥ ✐♥❡q✉❛❧✐t✐❡s ❜❡t✇❡❡♥

t❤❡ ❡r❣♦❞✐❝ s✉♠s ❛t t✐♠❡ qn✳ ❚❤❡♥✱ ❢♦r t❤❡ ❡r❣♦❞✐❝ s✉♠s ❣✐✈✐♥❣ ❛ ❜✐❣ ❡♥♦✉❣❤ ✈❛r✐❛♥❝❡✱ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡s❡ ❡r❣♦❞✐❝ s✉♠s ❛❢t❡r ♥♦r♠❛❧✐s❛t✐♦♥ t♦ t❤❡ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ❆ ♠❛✐♥ ♣♦✐♥t ✐s t♦ s❤♦✇ t❤❛t ❢♦r ❛ ❜✐❣ s❡t ♦❢ t✐♠❡s✱ t❤❡ ✈❛r✐❛♥❝❡ ✐s ❜✐❣ ❡♥♦✉❣❤✳ ❚❤✐s ✐s ✇❤❛t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❡①♣❧❛✐♥✳ ❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥ s❛t✐s✜❡❞ ❜② ❛✳❡✳ α ✿ ❍②♣♦t❤❡s✐s ✶✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝♦♥st❛♥ts A ≥ 1, p ≥ 0 s✉❝❤ t❤❛t an ≤ A np, ∀n ≥ 1. ✭✺✮

✾

❖♥❡ ✐♥❣r❡❞✐❡♥t ✐♥ t❤❡ ♣r♦♦❢ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ♦♥ t❤❡ ❞❡❝♦rr❡❧❛t✐♦♥ ❢♦r ❇❱ ❢✉♥❝t✐♦♥s ✿ Pr♦♣♦s✐t✐♦♥ ✶✳ ▲❡t ψ ❛♥❞ ϕ ❜❡ ❇❱ ❝❡♥t❡r❡❞ ❢✉♥❝t✐♦♥s✳ ❚❤❡♥✱ ✉♥✲ ❞❡r ❍②♣♦t❤❡s✐s ✶✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐♥❡q✉❛❧✐t✐❡s ❤♦❧❞ ❢♦r ❝♦♥st❛♥ts C, θ1, θ2, θ3✱ ❢♦r ❡✈❡r② 1 ≤ n ≤ m ≤ ℓ ✿ |

• X ψ ϕbnqn dµ| ≤ C V(ψ) V(ϕ) nθ1

qn bn, ✭✻✮ |

• X ψ ϕbnqnϕbmqm dµ| ≤ C V(ψ) V(ϕ)2 mθ2

qn bnbm, ✭✼✮ |

• X ψ ϕbnqnϕbmqmϕbℓqℓ dµ| ≤ C V(ψ) V(ϕ)3 ℓθ3

qn bnbmbℓ. ✭✽✮

✶✵

❆♥ ❛❜str❛❝t ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❋♦❧❧♦✇✐♥❣ t❤❡ ♠❡t❤♦❞ ✉s❡❞ ❜② ❍✉✈❡♥❡❡rs ✭✷✵✵✾✮ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ❞❡❝♦rr❡❧❛t✐♦♥ ❜♦✉♥❞✐♥❣ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡❝❡❞✐♥❣ ♣r♦♣♦s✐t✐♦♥ ♣❡r♠✐t t♦ ♦❜t❛✐♥ ❛ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞ ❛ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥✳ ❲❡ ❣✐✈❡ ❛ q✉❛♥t✐t❛t✐✈❡ ❢♦r♠✱ ❜♦✉♥❞✐♥❣ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ❖✉r ❣♦❛❧ ✐s t♦ ✇❡ ❣✐✈❡ ❝♦♥❞✐t✐♦♥s ✇❤✐❝❤ ✐♥s✉r❡ t❤❛t✱ ❢♦r ❡✈❡r② n✱ ❢♦r ❛ ✏❣♦♦❞✑ ♣r♦♣♦rt✐♦♥ ♦❢ ✐♥t❡❣❡rs k < qn✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ϕk/ϕk2 ✐s ❝❧♦s❡ t♦ ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✳ ❇✉t ✐t s❤♦✉❧❞ ❜❡ ♠❡♥t✐♦♥❡❞ t❤❛t t❤❡r❡ ❛r❡ ❛❧s♦ ❡①❛♠♣❧❡s ♦❢ r♦t❛t✐♦♥s ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ❛ ♥♦♥ ♥♦r♠❛❧ ❧✐♠✐t ❧❛✇ ❢♦r t❤❡ ❡r❣♦❞✐❝ s✉♠s ❛❧♦♥❣ t❤❡ s✉❜s❡q✉❡♥❝❡s ♦❢ ❜✐❣❣❡st ✈❛r✐❛♥❝❡✳ ❘❡❝❛❧❧ t❤❛t✱ ✐❢ X, Y ❛r❡ t✇♦ r✳r✳✈✳✬s ❞❡✜♥❡❞ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✱ t❤❡✐r ♠✉t✉❛❧ ❞✐st❛♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ✿ d(X, Y ) = supx∈R |P(X ≤ x)−P(Y ≤ x)|✳ ❇❡❧♦✇ Y1 ❞❡♥♦t❡s ❛ r✳✈✳ ✇✐t❤ ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ N(0, 1)✳ ▲❡t (qk)1≤k≤n ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs s❛t✐s✲ ❢②✐♥❣ t❤❡ ❧❛❝✉♥❛r② ❝♦♥❞✐t✐♦♥ ✿ t❤❡r❡ ❡①✐sts ρ < 1 s✉❝❤ t❤❛t qk/qm ≤ C ρm−k, 1 ≤ k < m ≤ n. ✭✾✮

✶✶

Pr♦♣♦s✐t✐♦♥ ✷✳ ▲❡t (fk)1≤k≤n ❜❡ r❡❛❧ ❝❡♥t❡r❡❞ ❇❱ ❢✉♥❝t✐♦♥s ❛♥❞ uk ❝♦♥st❛♥ts s✉❝❤ t❤❛t fk∞ ≤ uk ❛♥❞ V(fk) ≤ C uk qk, 1 ≤ k ≤ n,✳ ❙✉♣♣♦s❡ t❤❛t✱ ❢♦r ❛ ✜♥✐t❡ ❝♦♥st❛♥ts C > 0 ❛♥❞ θ ∈ R✱ ❢♦r ❡✈❡r② ❝❡♥t❡r❡❞ ❇❱ ❢✉♥❝t✐♦♥ ψ ✿ |

• X ψ fk dµ|

≤ C V(ψ) uk kθ qk , 1 ≤ k ≤ n, |

• X ψ fk fm dµ|

≤ C V (ψ) uk um mθ qk , 1 ≤ k ≤ m ≤ n, |

• X ψ fk fm fℓ dµ|

≤ C V (ψ) uk um uℓ ℓθ qk , 1 ≤ k ≤ m ≤ ℓ ≤ n. ❚❤❡♥✱ ❢♦r ❡✈❡r② δ > 0✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t C(δ) > 0 s✉❝❤ t❤❛t d( f1 + · · · + fn f1 + · · · + fn2 , Y1) ≤ C(δ) ( maxn

j=1 uj

f1 + · · · + fn2 )

2 3 n 1 4+δ.

✭✶✵✮

✶✷

❲❡ ❛♣♣❧② t❤✐s ❛❜str❛❝t ♣r♦♣♦s✐t✐♦♥ t♦ t❤❡ ❡r❣♦❞✐❝ s✉♠s ❢♦r ❛♥ ✐r✲ r❛t✐♦♥❛❧ r♦t❛t✐♦♥ ✿ t❤❡ qk✬s ❛r❡ t❤❡♥ t❤❡ ❞❡♥♦♠✐♥❛t♦rs ♦❢ α✱ ✇❤✐❝❤ s❛t✐s❢② ✭✾✮✱ ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥s fk ❛r❡ t❤❡ ❡r❣♦❞✐❝ s✉♠s ϕbkqk ♦❢ ❛ ❢✉♥❝t✐♦♥ ϕ ✭❝♦♠♣♦s❡❞ ❜② ❛ tr❛♥s❧❛t✐♦♥✮✱ ✇❤❡r❡ t❤❡ bk✬s ✭bk ≤ ak+1✮ ❛r❡ ❣✐✈❡♥ ❜② t❤❡ ❖str♦✇s❦✐✬s ❡①♣❛♥s✐♦♥ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ❋♦r t❤✐s r❡s✉❧t t♦ ❜❡ ✐♥t❡r❡st✐♥❣✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❤❛✈❡ ❛♥ ✐♥❢♦r♠❛✲ t✐♦♥ ❛❜♦✉t t❤❡ q✉♦t✐❡♥t

maxn

j=1 uj

f1+···+fn2✳ ❚❤✐s q✉❡st✐♦♥ ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞

❜❡❧♦✇ ❢♦r ❝♦♥❝r❡t❡ ❡①❛♠♣❧❡s✳ ■❢ fk∞ ≤ Ckτ, ∀k✱ ❛♥❞ ✐❢ f1 + · · · + fn2 ≥ c n

1 2+χ✱ ✇❤❡r❡ τ ❛♥❞ χ

❛r❡ t✇♦ ♣❛r❛♠❡t❡rs ❛♥❞ c ✐s ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✱ t❤❡♥ ✭✶✵✮ ✐♠♣❧✐❡s d( f1 + · · · + fn f1 + · · · + fn2 , Y1) ≤ C(δ) n− 1

12+2 3(τ−χ)+δ.

❋♦r ❡①❛♠♣❧❡✱ ✐❢ fk∞ ≤ C ❛♥❞ f1 + · · · + fn2 ≥ c n

1 2✱ t❤❡♥

d( f1 + · · · + fn f1 + · · · + fn2 , Y1) ≤ C(δ) n− 1

12+δ. ✶✸

❇♦✉♥❞s ❢♦r t❤❡ ✈❛r✐❛♥❝❡ ❢♦r ❛ ❧❛r❣❡ s❡t ♦❢ ✐♥t❡❣❡rs ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ϕ s❛t✐s✜❡s t❤❡ ❝♦♥❞✐t✐♦♥ ✿ ∃N0, η, θ0 > 0 s✳t✳ 1 N #{j ≤ N : |γqj(ϕ)| ≥ η} ≥ θ0, ∀N ≥ N0. ✭✶✶✮ ❚❤✐s ❝♦♥❞✐t✐♦♥ ✐s ❝❧❡❛r❧② s❛t✐s✜❡❞ ❜② ϕ(x) = {x}−1

2✱ s✐♥❝❡ ✐♥ t❤✐s ❝❛s❡

|γqj(ϕ)| = 1

2π, ∀j✳ ❚❤❡ ✈❛❧✐❞✐t② ♦❢ ✭✶✶✮ ❢♦r ❞✐✛❡r❡♥t st❡♣ ❢✉♥❝t✐♦♥s ✐s

❞✐s❝✉ss❡❞ ❧❛t❡r✳ ❚❤❡♦r❡♠ ✸✳ ■❢ α ✐s ❛ q✉❛❞r❛t✐❝ ♥✉♠❜❡r ❛♥❞ ϕ s❛t✐s✜❡s ❈♦♥❞✐t✐♦♥ ✭✶✶✮✱ t❤❡r❡ ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts ζ, η1, η2, R s✉❝❤ t❤❛t✱ ❢♦r N ❜✐❣ ❡♥♦✉❣❤ ✿ #{n ≤ N : η1 ln n ≤ ϕn2

2 ≤ η2 ln n} ≥ N (1 − R N−ζ).

✭✶✷✮ Pr♦♣♦s✐t✐♦♥ ✹✳ ■❢ ϕ s❛t✐s✜❡s ❈♦♥❞✐t✐♦♥ ✭✶✶✮✱ t❤❡♥ t❤❡ s✉❜s❡t W := {n ∈ N : ϕn2 ≥ c1

• m(n)/
• ln m(n)}

✭✶✸✮ s❛t✐s✜❡s lim

N→∞

#

• W ∩ [0, N[
• N

= 1. ✭✶✹✮

✶✹

❈▲❚ ✇✐t❤ r❛t❡ ❛❧♦♥❣ ❧❛r❣❡ s✉❜s❡ts ♦❢ ✐♥t❡❣❡rs ◆♦✇ ✇❡ st❛t❡ t❤❡ r❡s✉❧t ✐♥ ❛♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r ♦❢ ❣❡♥❡r❛❧✐t② ❢♦r α ❛♥❞ ❞❡❝r❡❛s✐♥❣ str❡♥❣t❤ ❢♦r t❤❡ ❝♦♥❝❧✉s✐♦♥✳ ▲❡t ϕ ❜❡ ❛ ❢✉♥❝t✐♦♥ ✐♥ C s❛t✐s❢②✐♥❣ ❈♦♥❞✐t✐♦♥ ✭✶✶✮✳ ❚❤❡♦r❡♠ ✺✳ ✶✮ ▲❡t α ❜❡ ❛ q✉❛❞r❛t✐❝ ✐rr❛t✐♦♥❛❧✳ ❋♦r t✇♦ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts 0 < b < B✱ ❧❡t Vb,B := {n ≥ 1 : b

• log n ≤ ϕn2 ≤ B
• log n}.

❚❤❡♥✱ t❤❡r❡ ❛r❡ b, B ❛♥❞ N0 ❛♥❞ t✇♦ ❝♦♥st❛♥ts R, ζ > 0 s✉❝❤ t❤❛t ✲ t❤❡ ❞❡♥s✐t② ♦❢ Vb s❛t✐s✜❡s ✿ Card(Vb,B

• [1, N]) ≥ N (1 − R N−ζ), ❢♦r N ≥ N0;

✭✶✺✮ ✲ ❢♦r δ0 ∈]0, 1

2[✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t K(δ0) s✉❝❤ t❤❛t ❢♦r n ∈ Vb,B✱

d( ϕn ϕn2 , Y1) ≤ K(δ0) (log n)− 1

12+δ0.

✭✶✻✮

✶✺

✷✮ ❙✉♣♣♦s❡ t❤❛t α ✐s ❜♣q✳ ❋♦r t✇♦ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts 0 < b < B✱ ❧❡t Wb,B := {n ≥ 1 : b (log log n)−1

2

• log n ≤ ϕn2 ≤ B
• log n}.

❚❤❡♥✱ t❤❡r❡ ❛r❡ b, B s✉❝❤ t❤❛t ✲ Wb,B ❤❛s ❞❡♥s✐t② ✶ ✐♥ t❤❡ ✐♥t❡❣❡rs ❀ ✲ ❢♦r δ0 ∈]0, 1

2[✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t K(δ0) s✉❝❤ t❤❛t ✭✶✻✮ ❤♦❧❞s ❢♦r

n ∈ Wb,B✳ ✸✮ ❙✉♣♣♦s❡ t❤❛t α ✐s s✉❝❤ t❤❛t an ≤ Cnp, ∀n ≥ 1✱ ❢♦r ❛ ❝♦♥st❛♥t C✳ ❋♦r t✇♦ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts 0 < b < B✱ ✇✐t❤ ♥♦t❛t✐♦♥ ✭✸✮ ❧❡t Zb,B := {n ∈ N : b

• m(n)/
• ln m(n) ≤ ϕn2 ≤ B
• m(n)}.

✭✶✼✮ ❚❤❡r❡ ❛r❡ b, B s✉❝❤ t❤❛t ✲ Zb,B ❤❛s ❞❡♥s✐t② ✶ ✐♥ t❤❡ ✐♥t❡❣❡rs ❀ ✲ ❢♦r δ0 ∈]0, 1

2[✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t K(δ0) s✉❝❤ t❤❛t✱ ✐❢ p < 1 8✱

d( ϕn ϕn2 , Y1) ≤ K(δ0) m(n)− 1

12+2 3p+δ0, ∀n ∈ Zb,B.

✭✶✽✮

✶✻

❆♣♣❧✐❝❛t✐♦♥ t♦ st❡♣ ❢✉♥❝t✐♦♥s ■❢ ϕ ❜❡❧♦♥❣s t♦ t❤❡ ❝❧❛ss C ♦❢ ❝❡♥t❡r❡❞ ❇❱ ❢✉♥❝t✐♦♥s✱ ✇✐t❤ ❋♦✉r✐❡r s❡r✐❡s

r=0 γr(ϕ) r

e2πir.✱ t♦ ❜❡ ❛❜❧❡ t♦ ❛♣♣❧② t❤❡ r❡s✉❧ts ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✶✮ ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts γqk(ϕ)✳ ❘❡❝❛❧❧ t❤❛t ✐t r❡❛❞s ✿ ∃N0, η, θ0 > 0 s✉❝❤ t❤❛t 1 N Card{j ≤ N : |γqj(ϕ)| ≥ η} ≥ θ0, ∀N ≥ N0. ❚❤❡ ❢✉♥❝t✐♦♥s {x} − 1

2 = −1 2πi

• r=0 1

r e2πirx ❛♥❞ 1[0,1

2[ − 1[1 2,1[ =

• r

2 πi(2r+1) e2πi(2r+1). ❛r❡ ✐♠♠❡❞✐❛t❡ ❡①❛♠♣❧❡s ✇❤❡r❡ ✭✶✶✮ ❛r❡ s❛t✐s✲

✜❡❞✳ ■♥ t❤❡ s❡❝♦♥❞ ❝❛s❡✱ ♦♥❡ ♦❜s❡r✈❡s t❤❛t γqk = 0 ✐❢ qk ✐s ❡✈❡♥, =

2 πi ✐❢ qk ✐s ♦❞❞✳ ❈❧❡❛r❧②✱ ✭✶✶✮ ✐s s❛t✐s✜❡❞ ❜❡❝❛✉s❡ t✇♦ ❝♦♥s❡❝✉t✐✈❡ qk✬s

❛r❡ r❡❧❛t✐✈❡❧② ♣r✐♠❡ ❛♥❞ t❤❡r❡❢♦r❡ ❝❛♥♥♦t ❜❡ ❜♦t❤ ❡✈❡♥✳

✶✼

■♥ ❣❡♥❡r❛❧✱ ❢♦r ❛ st❡♣ ❢✉♥❝t✐♦♥✱ ✭✶✶✮ ❛♥❞ t❤❡r❡❢♦r❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ ✈❛r✐❛♥❝❡ ϕn2

2 ❢♦r ♠❛♥② n✬s ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ❉✐♦♣❤❛♥t✐♥❡

♣r♦♣❡rt✐❡s ♦❢ ✐ts ❞✐s❝♦♥t✐♥✉✐t✐❡s ✇✐t❤ r❡s♣❡❝t t♦ α✳ ❲❡ ❞✐s❝✉ss ♥♦✇ t❤❡ ✈❛❧✐❞✐t② ♦❢ ✭✶✶✮✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❝❡♥t❡r❡❞ st❡♣ ❢✉♥❝t✐♦♥ ϕ ♦♥ [0, 1[ t❛❦✐♥❣ ❛ ❝♦♥st❛♥t ✈❛❧✉❡ vj ∈ R ♦♥ t❤❡ ✐♥t❡r✈❛❧ [uj, uj+1[✱ ✇✐t❤ u0 = 0 < u1 < ... < us < us+1 = 1 ✿ ϕ =

s

• j=0

vj 1[uj,uj+1[ − c. ✭✶✾✮ ❚❤❡ ❝♦♥st❛♥t c ❛❜♦✈❡ ✐s s✉❝❤ t❤❛t ϕ ✐s ❝❡♥t❡r❡❞✳ ▲❡♠♠❛ ✻✳ ■❢ ϕ ✐s ❣✐✈❡♥ ❜② ✭✶✾✮✱ t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥ Hϕ(u1, ..., us) ≥ 0 s✉❝❤ t❤❛t |γr(ϕ)|2 = π−2 Hϕ(ru1, ..., rus). ✭✷✵✮ ❊①❛♠♣❧❡ ✶ ✿ ϕ = ϕ(u, · ) = 1[0,u[ − u✱ Hϕ(u) = sin2(πu)✳ ❊①❛♠♣❧❡ ✷ ✿ ϕ = ϕ(w, u, · ) = 1[0, u] − 1[w, u+w]✱ H(ϕ) = 4 sin2(πu) sin2(πw)✳

✶✽

❈♦r♦❧❧❛r② ✼✳ ❈♦♥❞✐t✐♦♥ ✭✶✶✮ ✐s s❛t✐s✜❡❞ ✐❢ t❤❡ ♣❛r❛♠❡t❡r (u1, ..., us) ✐s s✉❝❤ t❤❛t t❤❡ s❡q✉❡♥❝❡ (qku1, ..., qkus)k≥1 ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ✐♥ Ts✳ ❙✐♥❝❡ (qk) ✐s ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡❣❡rs✱ ❢♦r ❛❧♠♦st ❡✈❡r② (u1, ..., us) ✐♥ Ts✱ t❤❡ s❡q✉❡♥❝❡ (qku1, ..., qkus)k≥1 ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ✐♥ Ts✳ ❍❡♥❝❡✱ ❢♦r ❛✳❡✳ (u1, ..., us) ∈ Ts ✿ lim

n

1 N

n

• k=1

Hϕ(qku1, ..., qkus) =

• Ts H(u1, ..., us)du1...dus > 0.

❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♥❡r✐❝ ♣r♦♣❡rt② ✿ ❈♦r♦❧❧❛r② ✽✳ ❈♦♥❞✐t✐♦♥ ✭✶✶✮ ✐s s❛t✐s✜❡❞ ❢♦r ❛✳❡✳ ✈❛❧✉❡ ♦❢ (u1, ..., us) ✐♥ Ts✳

✶✾

❇❡s✐❞❡s ❛ ❣❡♥❡r✐❝ r❡s✉❧t✱ t❤❡r❡ ❛r❡ ❛❧s♦ s♣❡❝✐✜❝ ✈❛❧✉❡s ♦❢ t❤❡ ♣❛r❛✲ ♠❡t❡r (u1, ..., us) ❢♦r ✇❤✐❝❤ ✭✶✶✮ ❤♦❧❞s✳ ❆ s✐♠♣❧❡ ❡①❛♠♣❧❡ ✐s ✿ ❊①❛♠♣❧❡ ✸ ✿ ϕ(r

s, · ) = 1[0,r

s[ − r

s✱ ❢♦r r, s ∈ N, 0 < r < s✳

❘❡♠❛r❦ ✶ ✿ ■♥ ❡①❛♠♣❧❡ ✶✱ ✐t ✐s ❦♥♦✇♥ t❤❛t ✐❢ α ✐s ❜♣q ❛♥❞ ✐❢ limk sin2(πqku) = 0✱ t❤❡♥ u ∈ Zα + Z✳ ❇✉t t❤❡r❡ ✐s ❛♥ ✉♥❝♦✉♥t❛❜❧❡ s❡t ♦❢ u✬s s✉❝❤ t❤❛t limN 1

N

N

k=1 sin2(πqku) = 0✳

❖❜s❡r✈❡ ❛❧s♦ t❤❛t✱ ✐❢ α ✐s ♥♦t ❜♣q✱ t❤❡r❡ ❛r❡ ♠❛♥② β✬s ✇❤✐❝❤ ❞♦ ♥♦t s❛t✐s❢② t❤❡ ♣r❡✈✐♦✉s ❡q✉✐❞✐str✐❜✉t✐♦♥ ♣r♦♣❡rt②✳ ▲❡t β =

n≥0 bnqnα ♠♦❞ 1, b

Z✱ ❜❡ t❤❡ s♦✲❝❛❧❧❡❞ ❖str♦✇s❦✐ ❡①♣❛♥s✐♦♥ ♦❢ β✱ ✇❤❡r❡ qn ❛r❡ t❤❡ ❞❡✲ ♥♦♠✐♥❛t♦rs ♦❢ α✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t✱ ✐❢

n≥0 |bn| an+1 < ∞✱ t❤❡♥

limk qkβ = 0✳ ❚❤❡r❡ ✐s ❛♥ ✉♥❝♦✉♥t❛❜❧❡ s❡t ♦❢ β✬s s❛t✐s❢②✐♥❣ t❤✐s ❝♦♥❞✐t✐♦♥ ✐❢ α ✐s ♥♦t ❜♣q✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ ❞❡❣❡♥❡r❛❝② ♦❢ t❤❡ ✈❛✲ r✐❛♥❝❡ ❢♦r t❤❡s❡ β✬s ✱ ❛❧t❤♦✉❣❤ ❡r❣♦❞✐❝✐t② ♦❢ t❤❡ ❝♦❝②❝❧❡ ❤♦❧❞s ✐❢ β ✐s ♥♦t ✐♥ t❤❡ ❝♦✉♥t❛❜❧❡ s❡t Zα + Z✳

✷✵

❱❡❝t♦r✐❛❧ ❝❛s❡ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ t✇♦ ❝♦♠♣♦♥❡♥ts✳ ▲❡t ❜❡ ❣✐✈❡♥ ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ Φ = (ϕ1, ϕ2)✱ ✇❤❡r❡ ϕ1, ϕ2 ❛r❡ t✇♦ ❝❡♥✲ t❡r❡❞ st❡♣ ❢✉♥❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t✐✈❡❧② s1✱ s2 ❞✐s❝♦♥t✐♥✉✐t✐❡s ✿ ϕi =

si

j=0 vi j 1[ui

j,ui j+1[ − ci✱ ❢♦r i = 1, 2✳

▲❡t t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Γn ❜❡ ❞❡✜♥❡❞ ❜② Γn(a, b) := (log n)−1aϕ1

n+

bϕ2

n2 2 ❛♥❞ I2 t❤❡ ✷✲❞✐♠❡♥s✐♦♥❛❧ ✐❞❡♥t✐t② ♠❛tr✐①✳

❚❤❡♦r❡♠ ✾✳ ■❢ α ✐s ❜♣q ❛♥❞ ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✶✮ ✐s s❛t✐s✜❡❞ ✉♥✐✲ ❢♦r♠❧② ✇✐t❤ r❡s♣❡❝t t♦ (a, b) ✐♥ t❤❡ ✉♥✐t s♣❤❡r❡✱ t❤❡r❡ ❛r❡ 0 < r1, r2 < +∞ t✇♦ ❝♦♥st❛♥ts s✉❝❤ t❤❛t ❢♦r ❛ ✏❧❛r❣❡✑ s❡t ♦❢ ✐♥t❡❣❡rs n ❛s ❛❜♦✈❡ ✿ ✲ Γn s❛t✐s✜❡s ✐♥❡q✉❛❧✐t✐❡s ♦❢ t❤❡ ❢♦r♠ r1I2 ≤ Γn(a, b) ≤ r1I2 ❀ ✲ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Γ−1

n Φn ✐s ❝❧♦s❡ t♦ t❤❡ st❛♥❞❛r❞ ✷✲❞✐♠❡♥s✐♦♥❛❧

♥♦r♠❛❧ ❧❛✇✳

✷✶

■♥ ♦r❞❡r t♦ ❛♣♣❧② t❤❡ t❤❡♦r❡♠✱ ✇❡ ♥❡❡❞ ✭✶✶✮ ✉♥✐❢♦r♠❧② ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❚♦ ❣❡t ✐t✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✶✵✳ ▲❡t Λ ❜❡ ❛ ❝♦♠♣❛❝t s♣❛❝❡ ❛♥❞ (Fλ, λ ∈ Λ) ❜❡ ❛ ❢❛♠✐❧② ♦❢ ♥♦♥♥❡❣❛t✐✈❡ ♥♦♥ ✐❞❡♥t✐❝❛❧❧② ♥✉❧❧ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ Td ❞❡♣❡♥❞✐♥❣ ❝♦♥t✐♥✉♦✉s❧② ♦♥ λ✳ ■❢ ❛ s❡q✉❡♥❝❡ (zn) ✐s ❡q✉✐❞✐str✐❜✉t❡❞ ✐♥ Td✱ t❤❡♥ ∃ N0, η, η0 > 0 s✳ t✳ #{n ≤ N : Fλ(zn) ≥ η} ≥ θ N, ∀N ≥ N0, ∀λ ∈ Λ. ❆ ❣❡♥❡r✐❝ r❡s✉❧t ❇② Pr♦♣♦s✐t✐♦♥ ✶✵ ❛♣♣❧✐❡❞ ❢♦r (a, b) ✐♥ t❤❡ ✉♥✐t s♣❤❡r❡✱ ❢♦r ❛✳❡✳ ✈❛✲ ❧✉❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡r (u1

1, ..., u1 s1, u2 1, ..., u2 s2)✱ t❤❡ ❢✉♥❝t✐♦♥s aϕ1 +bϕ2

s❛t✐s❢② ❈♦♥❞✐t✐♦♥ ✭✶✶✮ ✉♥✐❢♦r♠❧② ✐♥ (a, b) ✐♥ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❍❡♥❝❡ ❚❤❡♦r❡♠ ✾ ❛♣♣❧✐❡s ❣❡♥❡r✐❝❛❧❧② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s✳

✷✷

❙♣❡❝✐❛❧ ✈❛❧✉❡s ✿ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ r❡❝t❛♥❣✉❧❛r ❜✐❧❧✐❛r❞ ✐♥ t❤❡ ♣❧❛♥❡ ❊①❛♠♣❧❡ ✹ ❈♦♥s✐❞❡r t❤❡ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❛♣♣❡❛rs ✐♥ t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ ♣❡r✐♦❞✐❝ ❜✐❧❧✐❛r❞ ✿ ψ = (ϕ1, ϕ2) ✇✐t❤ ϕ1 = 1[0,α

2] − 1[1 2,1 2+α 2], ϕ2 = 1[0,1 2−α 2] − 1[1 2,1−α 2].

❚❤❡ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥t ♦❢ ϕ1 ❛♥❞ ϕ2 ♦❢ ♦r❞❡r r ❛r❡ ♥✉❧❧ ❢♦r r ❡✈❡♥✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ϕa,b = aϕ1 + bϕ2✳ ■❢ qj ✐s ❡✈❡♥✱ γqj(ϕa,b) ✐s ♥✉❧❧✳ ■❢ qj ✐s ♦❞❞✱ ✇❡ ❤❛✈❡ γqj(ϕa,b) = a(1 + O(

1 qj+1))✱ ✐❢ pj ✐s ♦❞❞✱ = b(1 + O( 1 qj+1))✱ ✐❢ pj ✐s

❡✈❡♥✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ s❤♦✇s t❤❛t✱ ✐❢ α ✐s s✉❝❤ t❤❛t✱ ✐♥ ❛✈❡r❛❣❡✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ♣r♦♣♦rt✐♦♥ ♦❢ ♣❛✐rs (pj, qj) ✭❡✈❡♥✱ ♦❞❞✮ ❛♥❞ ❛ ♣♦s✐t✐✈❡ ♣r♦✲ ♣♦rt✐♦♥ ♦❢ ♣❛✐rs (pj, qj) ✭♦❞❞✱ ♦❞❞✮✱ t❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✾ ✐s ❢✉❧✜❧❧❡❞ ❜② t❤❡ ✈❡❝t♦r✐❛❧ st❡♣ ❢✉♥❝t✐♦♥ ψ = (ϕ1, ϕ2)✳

✷✸

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥

✷✹

✳ ✳ ❆♥ ♦r❜✐t ♦❢ t❤❡ r❡❝t❛♥❣✉❧❛r ❜✐❧❧✐❛r❞✱ ❛♥❣❧❡ π/4

✶

✳ ✳ ❚❤❡ ❜✐❧❧✐❛r❞ t❛❜❧❡✱ ❛♥❣❧❡ π/4

✷