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P❡r❝♦❧❛t✐♦♥ ❛♥❞ r❛♥❞♦♠ ♥♦❞❛❧ ❧✐♥❡s

❘❛♥❞♦♠ ✇❛✈❡s ✐♥ ❖①❢♦r❞✲ ✶✽✲✷✷ ❏✉♥❡ ✷✵✶✽

❉❛♠✐❡♥ ●❛②❡t ✭■♥st✐t✉t ❋♦✉r✐❡r✱ ●r❡♥♦❜❧❡✮ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱✐♥❝❡♥t ❇❡✛❛r❛ ✭■♥st✐t✉t ❋♦✉r✐❡r✱ ●r❡♥♦❜❧❡✮

✶✴✺✶

lim inf

n,m→∞ Pr♦❜ > c > 0?

✷✴✺✶

Pr♦❜ →

n,λ→∞ 0

✸✴✺✶

Pr♦❜ →

n,λ→∞ 1

✹✴✺✶

lim inf

n→∞ Pr♦❜ ≥ c > 0 ?

✺✴✺✶

❙q✉❛r❡s

❲✐t❤ s②♠♠❡tr② ❜❡t✇❡❡♥ ✰ ❛♥❞ ✲ s②♠♠❡tr② ❜❡t✇❡❡♥ ❛♥❞ t❤❡♥ ❜♦t❤ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡q✉❛❧✳✳✳

✻✴✺✶

❙q✉❛r❡s

❲✐t❤

◮ s②♠♠❡tr② ❜❡t✇❡❡♥ ✰ ❛♥❞ ✲ ◮ s②♠♠❡tr② ❜❡t✇❡❡♥ x1 ❛♥❞ x2

t❤❡♥ ❜♦t❤ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡q✉❛❧✳✳✳

✻✴✺✶

Pr♦❜ = 1/2.

✼✴✺✶

❇♦♥❞ ♣❡r❝♦❧❛t✐♦♥ ♦♥ Z2✳ ❚❤❡♦r❡♠ ✭❘✉ss♦✱ ❙❡②♠♦✉r✲❲❡❧s❤ ✶✾✼✽✮ ▲❡t R ⊂ R2✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts c > 0✱ lim inf

n→∞ Pr♦❜ (♣♦s✐t✐✈❡ ❝r♦ss✐♥❣ ♦❢ nR) > c.

✽✴✺✶

◗✉❡st✐♦♥✿ ▲❡t f : R2 → R ❛ ❜❡ r❛♥❞♦♠ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ❛♥❞ ✜① R ⊂ R2✳ ❉♦❡s ✐t ❡①✐st c > 0✱ lim inf

n→∞ Pr♦❜

• {f > 0} ❝r♦ss❡s nR
• > c?

✾✴✺✶

▲❡t f : R2 → R ❜❡

◮ ❛ ❝❡♥t❡r❡❞ ●❛✉ss✐❛♥ ✜❡❧❞ ◮ ✇✐t❤ s②♠♠❡tr✐❝ ❝♦✈❛r✐❛♥t ❢✉♥❝t✐♦♥

e(x, y) := E(f(x)f(y)) = k(x − y). ❚✇♦ ✉♥✐✈❡rs❛❧ ♠♦❞❡❧s ❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧ ✭❘❲✮ ✭❘✐❡♠❛♥♥✐❛♥✮ ❚❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ ♠♦❞❡❧ ✭❛❧❣❡❜r❛✐❝✮

✶✵✴✺✶

▲❡t f : R2 → R ❜❡

◮ ❛ ❝❡♥t❡r❡❞ ●❛✉ss✐❛♥ ✜❡❧❞ ◮ ✇✐t❤ s②♠♠❡tr✐❝ ❝♦✈❛r✐❛♥t ❢✉♥❝t✐♦♥

e(x, y) := E(f(x)f(y)) = k(x − y). ❚✇♦ ✉♥✐✈❡rs❛❧ ♠♦❞❡❧s

◮ ❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧ ✭❘❲✮ ✭❘✐❡♠❛♥♥✐❛♥✮ ◮ ❚❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ ♠♦❞❡❧ ✭❛❧❣❡❜r❛✐❝✮

✶✵✴✺✶

❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧

❇❛r♥❡tt✱ ❇♦❣♦♠♦❧♥②✲❙❝❤♠✐❞t

◮ g(r, θ) = ∞ m=−∞ amJ|m|(r)eimθ

❧✐♠✐t ♠♦❞❡❧ ❢♦r t❤❡ r❡s❝❛❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s ✭❛♥❞ ♠♦r❡ ✲ ✉♥✐✈❡rs❛❧ ❢r♦♠ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s✮✳

✶✶✴✺✶

❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧

❇❛r♥❡tt✱ ❇♦❣♦♠♦❧♥②✲❙❝❤♠✐❞t

◮ g(r, θ) = ∞ m=−∞ amJ|m|(r)eimθ ◮ ❧✐♠✐t ♠♦❞❡❧ ❢♦r t❤❡ r❡s❝❛❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s ◮ ✭❛♥❞ ♠♦r❡ ✲ ✉♥✐✈❡rs❛❧ ❢r♦♠ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s✮✳

✶✶✴✺✶

❈♦♥❥❡❝t✉r❡ ✭❇♦❣♦♠♦❧♥②✲❙❝❤♠✐❞t ✷✵✵✼✮ ❘❙❲ ❢♦r t❤✐s ♠♦❞❡❧✳

✶✷✴✺✶

❚❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ ♠♦❞❡❧

◆❛st❛s❡s❝✉ ✲ ❇❡✛❛r❛

◮ f(x1, x2) = ∞ i,j=0 aij xi

1xj 2

√i!j!

✐s t❤❡ ❧✐♠✐t ❢♦r t❤❡ r❡s❝❛❧❡❞ ♣♦❧②♥♦♠✐❛❧s ❢♦r ❝♦♠♣❧❡① ❋✉❜✐♥✐✲❙t✉❞② ✭❑♦st❧❛♥✮ ♠❡❛s✉r❡✳ ✭❛♥❞ ♠♦r❡ ✲ ✉♥✐✈❡rs❛❧ ❢r♦♠ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t✐❡s✮✳

✶✸✴✺✶

❚❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ ♠♦❞❡❧

◆❛st❛s❡s❝✉ ✲ ❇❡✛❛r❛

◮ f(x1, x2) = ∞ i,j=0 aij xi

1xj 2

√i!j! ◮ ✐s t❤❡ ❧✐♠✐t ❢♦r t❤❡ r❡s❝❛❧❡❞ ♣♦❧②♥♦♠✐❛❧s ❢♦r ❝♦♠♣❧❡①

❋✉❜✐♥✐✲❙t✉❞② ✭❑♦st❧❛♥✮ ♠❡❛s✉r❡✳

◮ ✭❛♥❞ ♠♦r❡ ✲ ✉♥✐✈❡rs❛❧ ❢r♦♠ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t✐❡s✮✳

✶✸✴✺✶

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✻✮ ❘❙❲ ❤♦❧❞s ❢♦r ❇❛r❣♠❛♥♥✲❋♦❝❦✿ ❢♦r ❛♥② r❡❝t❛♥❣❧❡ R✱ t❤❡r❡ ❡①✐sts c > 0 s✉❝❤ t❤❛t lim inf

n→∞ Pr♦❜

• {f > 0} ❝r♦ss❡s nR
• > c.

✶✹✴✺✶

❘❡♠❛r❦✿ ❘❙❲ ❤♦❧❞s ❢♦r 0 ≤ k(x − y) ≤ x − y−325

◮ ❇❡❧②❛❡✈✲▼✉✐r❤❡❛❞✿ 325 → 16 ◮ ❘✐✈❡r❛✲❱❛♥♥❡✉✈✐❧❧❡✿ 325 → 4✳

✶✺✴✺✶

❈♦r♦❧❧❛r② ✭❇❡✛❛r❛✲●✮ ❋♦r ❇❛r❣♠❛♥♥✲❋♦❝❦✱ Pr♦❜ < ℓ n α>0

✶✻✴✺✶

❈♦r♦❧❧❛r② ✭❆❧❡①❛♥❞❡r ✶✾✾✻✮ ❆❧♠♦st s✉r❡❧② t❤❡r❡ ✐s ♥♦ ✐♥✜♥✐t❡ ❝♦♠♣♦♥❡♥t ♦❢ {f > 0}✳ ❚❤❡♦r❡♠ ✭❘✐✈❡r❛✲❱❛♥♥❡✉✈✐❧❧❡ ✷✵✶✼✮ ❋♦r ❛♥② ✱ ❛❧♠♦st s✉r❡❧② ❛s ❛♥ ✐♥✜♥✐t❡ ❝♦♠♣♦♥❡♥t✳

✶✼✴✺✶

❈♦r♦❧❧❛r② ✭❆❧❡①❛♥❞❡r ✶✾✾✻✮ ❆❧♠♦st s✉r❡❧② t❤❡r❡ ✐s ♥♦ ✐♥✜♥✐t❡ ❝♦♠♣♦♥❡♥t ♦❢ {f > 0}✳ ❚❤❡♦r❡♠ ✭❘✐✈❡r❛✲❱❛♥♥❡✉✈✐❧❧❡ ✷✵✶✼✮ ❋♦r ❛♥② ǫ > 0✱ ❛❧♠♦st s✉r❡❧② {f > −ǫ} ❛s ❛♥ ✐♥✜♥✐t❡ ❝♦♠♣♦♥❡♥t✳

✶✼✴✺✶

❚❤❡♦r❡♠ ✭❇❡❧②❛❡✈✲▼✉✐r❤❡❛❞✲❲✐❣♠❛♥ ✷✵✶✼✮ ❘❙❲ ❤♦❧❞s ❢♦r ♣♦❧②♥♦♠✐❛❧s ✇✐t❤ t❤❡ ❋✉❜✐♥✐✲❙t✉❞✐ ♠❡❛s✉r❡✳

✶✽✴✺✶

❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄

◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿

e(x, y) = exp(−x − y2).

✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡

❘❛♥❞♦♠ ✇❛✈❡s✿

✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② str♦♥❣ ❞❡♣❡♥❞❡♥❝❡

✶✾✴✺✶

❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄

◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿

e(x, y) = exp(−x − y2).

✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② → ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡

❘❛♥❞♦♠ ✇❛✈❡s✿

✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② str♦♥❣ ❞❡♣❡♥❞❡♥❝❡

✶✾✴✺✶

❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄

◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿

e(x, y) = exp(−x − y2).

✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② → ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡

◮ ❘❛♥❞♦♠ ✇❛✈❡s✿

e(x, y) = J0(x − y)

✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② str♦♥❣ ❞❡♣❡♥❞❡♥❝❡

✶✾✴✺✶

❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄

◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿

e(x, y) = exp(−x − y2).

✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② → ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡

◮ ❘❛♥❞♦♠ ✇❛✈❡s✿

e(x, y) = J0(x − y)

✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② → str♦♥❣ ❞❡♣❡♥❞❡♥❝❡

✶✾✴✺✶

❙tr♦♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐s ♥♦t ❡♥♦✉❣❤✳✳✳

✳✳✳ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❆♥❛❧②t✐❝ ❈♦♥t✐♥✉❛t✐♦♥ P❤❡♥♦♠❡♥♦♥✳

✷✵✴✺✶

❙tr♦♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐s ♥♦t ❡♥♦✉❣❤✳✳✳

✳✳✳ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❆♥❛❧②t✐❝ ❈♦♥t✐♥✉❛t✐♦♥ P❤❡♥♦♠❡♥♦♥✳

✷✵✴✺✶

❙tr♦♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐s ♥♦t ❡♥♦✉❣❤✳✳✳

✳✳✳ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❆♥❛❧②t✐❝ ❈♦♥t✐♥✉❛t✐♦♥ P❤❡♥♦♠❡♥♦♥✳

✷✵✴✺✶

❙♦❧✉t✐♦♥ ✿ ❜❧✉rr✐♥❣ ❜② ❞✐s❝r❡t✐③❛t✐♦♥

◮ T = tr✐❛♥❣✉❧❛r ❧❛tt✐❝❡✱ ◮ V = ✐ts ✈❡rt✐❝❡s✱ ◮ s✐❣♥ f|V : V → {±1}✳ ◮ ❙✐t❡ ♣❡r❝♦❧❛t✐♦♥✿ t❤❡ ❡❞❣❡ ✐s ♣♦s✐t✐✈❡ ✐✛ ✐ts ❡①tr❡♠✐t✐❡s ❛r❡✳

✷✶✴✺✶

■s t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ tr✉st❢✉❧❄

✶✳ ■❢ ✐s t♦♦ ❝♦❛rs❡✱ t❤❡♥ ♥♦✳ ✷✳ ■❢ ✐s ✈❡r② t❤✐♥✱ t❤❡♥ ②❡s✱ ❜✉t✳✳✳ ❞❡♣❡♥❞❡♥❝❡ ❝♦♠❡s ❜❛❝❦✳

✷✷✴✺✶

■s t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ tr✉st❢✉❧❄

✶✳ ■❢ T ✐s t♦♦ ❝♦❛rs❡✱ t❤❡♥ ♥♦✳ ✷✳ ■❢ ✐s ✈❡r② t❤✐♥✱ t❤❡♥ ②❡s✱ ❜✉t✳✳✳ ❞❡♣❡♥❞❡♥❝❡ ❝♦♠❡s ❜❛❝❦✳

✷✷✴✺✶

■s t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ tr✉st❢✉❧❄

✶✳ ■❢ T ✐s t♦♦ ❝♦❛rs❡✱ t❤❡♥ ♥♦✳ ✷✳ ■❢ T ✐s ✈❡r② t❤✐♥✱ t❤❡♥ ②❡s✱ ❜✉t✳✳✳ ❞❡♣❡♥❞❡♥❝❡ ❝♦♠❡s ❜❛❝❦✳

✷✷✴✺✶

■s t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ tr✉st❢✉❧❄

✶✳ ■❢ T ✐s t♦♦ ❝♦❛rs❡✱ t❤❡♥ ♥♦✳ ✷✳ ■❢ T ✐s ✈❡r② t❤✐♥✱ t❤❡♥ ②❡s✱ ❜✉t✳✳✳ ❞❡♣❡♥❞❡♥❝❡ ❝♦♠❡s ❜❛❝❦✳

✷✷✴✺✶

◗✉❛♥t✐t❛t✐✈❡ ❣♦♦❞ ❜❧✉rr✐♥❣

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✻✮ ■♥ [0, n]2✱ ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ ❝♦♥t✐♥✉♦✉s ❝r♦ss✐♥❣s ⇔ ❞✐s❝r❡t❡ ❝r♦ss✐♥❣s ✐♥ 1 n9 T .

✷✸✴✺✶

◗✉❛♥t✐t❛t✐✈❡ ❞❡♣❡♥❞❡♥❝❡

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✻ ✲ ❱✳ P✐t❡r❜❛r❣ ✶✾✽✷✮ max

A ❝r♦ss✐♥❣ ✐♥ nR A′ ❝r♦ss✐♥❣ ✐♥ nR′

|Pr♦❜ (A ❡t A′) − Pr♦❜ A Pr♦❜ A′| ≤ (# ✈❡rt✐❝❡s ✐♥ nR ❛♥❞ nR′)8/5 max

x∈nR y∈nR′

|e(x, y)|1/5. ❋♦r ♦✉r ❞✐s❝r❡t✐③❛t✐♦♥ s❝❤❡♠❡ ❢♦r ❇❛r❣♠❛♥♥✲❋♦❝❦✱ ♦♥ t✇♦ ❞✐s❥♦✐♥t ❛♥❞ ✱ t❤✐s ❣✐✈❡s ❞❡♣❡♥❞❡♥❝❡

✷✹✴✺✶

◗✉❛♥t✐t❛t✐✈❡ ❞❡♣❡♥❞❡♥❝❡

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✻ ✲ ❱✳ P✐t❡r❜❛r❣ ✶✾✽✷✮ max

A ❝r♦ss✐♥❣ ✐♥ nR A′ ❝r♦ss✐♥❣ ✐♥ nR′

|Pr♦❜ (A ❡t A′) − Pr♦❜ A Pr♦❜ A′| ≤ (# ✈❡rt✐❝❡s ✐♥ nR ❛♥❞ nR′)8/5 max

x∈nR y∈nR′

|e(x, y)|1/5. ❋♦r ♦✉r ❞✐s❝r❡t✐③❛t✐♦♥ s❝❤❡♠❡ ❢♦r ❇❛r❣♠❛♥♥✲❋♦❝❦✱ ♦♥ t✇♦ ❞✐s❥♦✐♥t R ❛♥❞ R′✱ t❤✐s ❣✐✈❡s ❞❡♣❡♥❞❡♥❝❡(nR, nR′) ≤ n50e−n2/5

✷✹✴✺✶

◗✉❛♥t✐t❛t✐✈❡ ❞❡♣❡♥❞❡♥❝❡

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✻ ✲ ❱✳ P✐t❡r❜❛r❣ ✶✾✽✷✮ max

A ❝r♦ss✐♥❣ ✐♥ nR A′ ❝r♦ss✐♥❣ ✐♥ nR′

|Pr♦❜ (A ❡t A′) − Pr♦❜ A Pr♦❜ A′| ≤ (# ✈❡rt✐❝❡s ✐♥ nR ❛♥❞ nR′)8/5 max

x∈nR y∈nR′

|e(x, y)|1/5. ❋♦r ♦✉r ❞✐s❝r❡t✐③❛t✐♦♥ s❝❤❡♠❡ ❢♦r ❇❛r❣♠❛♥♥✲❋♦❝❦✱ ♦♥ t✇♦ ❞✐s❥♦✐♥t R ❛♥❞ R′✱ t❤✐s ❣✐✈❡s ❞❡♣❡♥❞❡♥❝❡(nR, nR′) ≤ n50e−n2/5 →

n→∞ 0.

✷✹✴✺✶

❆ ❝r✉❝✐❛❧ t♦♦❧ ❢♦r ❘❙❲

❋❑● ✭❋♦rt✉✐♥✲❑❛st❡❧❡②♥✲●✐♥✐❜r❡✮ ✭❝r♦ss✐♥❣ ❂ ♣♦s✐t✐✈❡ ❝r♦ss✐♥❣✮✳ ❋❑● ✐♠♣❧✐❡s Pr♦❜

• ❝r♦ss✐♥❣ ♦❢ R

∩ ❝r♦ss✐♥❣ ♦❢ R′ ≥ Pr♦❜(❝r♦ss✐♥❣ ♦❢ R) . Pr♦❜(❝r♦ss✐♥❣ ♦❢ R′).

✷✺✴✺✶

Pr♦❜ ✭❝r♦ss✐♥❣ t❤❡ r❡❝t❛♥❣❧❡

✷✻✴✺✶

Pr♦❜ ✭❝r♦ss✐♥❣ t❤❡ r❡❝t❛♥❣❧❡

✷✻✴✺✶

= Pr♦❜ ✭❝r♦ss✐♥❣ t❤❡ r❡❝t❛♥❣❧❡)2

✷✻✴✺✶

Pr♦❜ ≥ Pr♦❜ ✭❝r♦ss✐♥❣ t❤❡ r❡❝t❛♥❣❧❡)4

✷✼✴✺✶

❚❤❡♦r❡♠ ✭▲♦r❡♥ P✐tt ✶✾✽✷✮ ❋♦r ●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥s✱ FKG ⇔ ♣♦s✐t✐✈❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥.

✷✽✴✺✶

❚❤❡♦r❡♠ ✭❚❛ss✐♦♥ ✷✵✶✻✮ ■❢ ✇❡ ❤❛✈❡ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s ✇✐t❤ ✶✳ ❋❑● ✷✳ ✉♥✐❢♦r♠ ❝r♦ss✐♥❣ ♦❢ sq✉❛r❡s ✸✳ ✉♥✐❢♦r♠ ❛s②♠♣t♦t✐❝ ✐♥❞❡♣❡♥❞❡♥❝❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ✉♥✐❢♦r♠❧② ♣♦s✐t✐✈❡❧② ❜♦✉♥❞❡❞ ❘❙❲✳ ■♥ ❝♦♥❝❧✉s✐♦♥✿ ❢♦r ❡✈❡r② ✇❡ ❞✐s❝r❡t✐③❡ ♦♥ ✇✐t❤ ❤✐❣❤ ✉♥✐❢♦r♠ ♣r♦❜❛❜✐❧✐t② t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡ ❝r♦ss✐♥❣s ❤❛♣♣❡♥ s✐♠✉❧t❛♥❡♦✉s❧② t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ s❛t✐s✜❡s t❤❡ t❤r❡❡ ❢♦r♠❡r ❝♦♥❞✐t✐♦♥s ✉♥✐❢♦r♠❧② ✐♥ ✳ ❚❤❡♥ ❚❛ss✐♦♥ ❣✐✈❡s ❛ ✉♥✐❢♦r♠ ❘❙❲ ❢♦r ❡✈❡r② s❝❛❧❡ ✳

✷✾✴✺✶

❚❤❡♦r❡♠ ✭❚❛ss✐♦♥ ✷✵✶✻✮ ■❢ ✇❡ ❤❛✈❡ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s ✇✐t❤ ✶✳ ❋❑● ✷✳ ✉♥✐❢♦r♠ ❝r♦ss✐♥❣ ♦❢ sq✉❛r❡s ✸✳ ✉♥✐❢♦r♠ ❛s②♠♣t♦t✐❝ ✐♥❞❡♣❡♥❞❡♥❝❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ✉♥✐❢♦r♠❧② ♣♦s✐t✐✈❡❧② ❜♦✉♥❞❡❞ ❘❙❲✳ ■♥ ❝♦♥❝❧✉s✐♦♥✿

◮ ❢♦r ❡✈❡r② n ✇❡ ❞✐s❝r❡t✐③❡ ♦♥ [0, n]2

✇✐t❤ ❤✐❣❤ ✉♥✐❢♦r♠ ♣r♦❜❛❜✐❧✐t② t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡ ❝r♦ss✐♥❣s ❤❛♣♣❡♥ s✐♠✉❧t❛♥❡♦✉s❧② t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ s❛t✐s✜❡s t❤❡ t❤r❡❡ ❢♦r♠❡r ❝♦♥❞✐t✐♦♥s ✉♥✐❢♦r♠❧② ✐♥ ✳ ❚❤❡♥ ❚❛ss✐♦♥ ❣✐✈❡s ❛ ✉♥✐❢♦r♠ ❘❙❲ ❢♦r ❡✈❡r② s❝❛❧❡ ✳

✷✾✴✺✶

❚❤❡♦r❡♠ ✭❚❛ss✐♦♥ ✷✵✶✻✮ ■❢ ✇❡ ❤❛✈❡ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s ✇✐t❤ ✶✳ ❋❑● ✷✳ ✉♥✐❢♦r♠ ❝r♦ss✐♥❣ ♦❢ sq✉❛r❡s ✸✳ ✉♥✐❢♦r♠ ❛s②♠♣t♦t✐❝ ✐♥❞❡♣❡♥❞❡♥❝❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ✉♥✐❢♦r♠❧② ♣♦s✐t✐✈❡❧② ❜♦✉♥❞❡❞ ❘❙❲✳ ■♥ ❝♦♥❝❧✉s✐♦♥✿

◮ ❢♦r ❡✈❡r② n ✇❡ ❞✐s❝r❡t✐③❡ ♦♥ [0, n]2 ◮ ✇✐t❤ ❤✐❣❤ ✉♥✐❢♦r♠ ♣r♦❜❛❜✐❧✐t② t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡

❝r♦ss✐♥❣s ❤❛♣♣❡♥ s✐♠✉❧t❛♥❡♦✉s❧② t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ s❛t✐s✜❡s t❤❡ t❤r❡❡ ❢♦r♠❡r ❝♦♥❞✐t✐♦♥s ✉♥✐❢♦r♠❧② ✐♥ ✳ ❚❤❡♥ ❚❛ss✐♦♥ ❣✐✈❡s ❛ ✉♥✐❢♦r♠ ❘❙❲ ❢♦r ❡✈❡r② s❝❛❧❡ ✳

✷✾✴✺✶

❚❤❡♦r❡♠ ✭❚❛ss✐♦♥ ✷✵✶✻✮ ■❢ ✇❡ ❤❛✈❡ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s ✇✐t❤ ✶✳ ❋❑● ✷✳ ✉♥✐❢♦r♠ ❝r♦ss✐♥❣ ♦❢ sq✉❛r❡s ✸✳ ✉♥✐❢♦r♠ ❛s②♠♣t♦t✐❝ ✐♥❞❡♣❡♥❞❡♥❝❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ✉♥✐❢♦r♠❧② ♣♦s✐t✐✈❡❧② ❜♦✉♥❞❡❞ ❘❙❲✳ ■♥ ❝♦♥❝❧✉s✐♦♥✿

◮ ❢♦r ❡✈❡r② n ✇❡ ❞✐s❝r❡t✐③❡ ♦♥ [0, n]2 ◮ ✇✐t❤ ❤✐❣❤ ✉♥✐❢♦r♠ ♣r♦❜❛❜✐❧✐t② t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡

❝r♦ss✐♥❣s ❤❛♣♣❡♥ s✐♠✉❧t❛♥❡♦✉s❧②

◮ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ s❛t✐s✜❡s t❤❡ t❤r❡❡ ❢♦r♠❡r ❝♦♥❞✐t✐♦♥s

✉♥✐❢♦r♠❧② ✐♥ n✳ ❚❤❡♥ ❚❛ss✐♦♥ ❣✐✈❡s ❛ ✉♥✐❢♦r♠ ❘❙❲ ❢♦r ❡✈❡r② s❝❛❧❡ ✳

✷✾✴✺✶

❚❤❡♦r❡♠ ✭❚❛ss✐♦♥ ✷✵✶✻✮ ■❢ ✇❡ ❤❛✈❡ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s ✇✐t❤ ✶✳ ❋❑● ✷✳ ✉♥✐❢♦r♠ ❝r♦ss✐♥❣ ♦❢ sq✉❛r❡s ✸✳ ✉♥✐❢♦r♠ ❛s②♠♣t♦t✐❝ ✐♥❞❡♣❡♥❞❡♥❝❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ✉♥✐❢♦r♠❧② ♣♦s✐t✐✈❡❧② ❜♦✉♥❞❡❞ ❘❙❲✳ ■♥ ❝♦♥❝❧✉s✐♦♥✿

◮ ❢♦r ❡✈❡r② n ✇❡ ❞✐s❝r❡t✐③❡ ♦♥ [0, n]2 ◮ ✇✐t❤ ❤✐❣❤ ✉♥✐❢♦r♠ ♣r♦❜❛❜✐❧✐t② t❤❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡

❝r♦ss✐♥❣s ❤❛♣♣❡♥ s✐♠✉❧t❛♥❡♦✉s❧②

◮ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ s❛t✐s✜❡s t❤❡ t❤r❡❡ ❢♦r♠❡r ❝♦♥❞✐t✐♦♥s

✉♥✐❢♦r♠❧② ✐♥ n✳

◮ ❚❤❡♥ ❚❛ss✐♦♥ ❣✐✈❡s ❛ ✉♥✐❢♦r♠ ❘❙❲ ❢♦r ❡✈❡r② s❝❛❧❡ n✳

✷✾✴✺✶

❲✐t❤♦✉t ♣♦s✐t✐✈❡ ❝♦rr❡❧❛t✐♦♥s ✭✇✐t❤♦✉t ❋❑●✮❄

◮ fB : V → R ●❛✉ss✐❛♥ ✜❡❧❞✱

s✐❣♥fB = ❇❡r♥♦✉❧❧✐

◮ f : V → R ●❛✉ss✐❛♥ ✜❡❧❞ ◮ s②♠♠❡tr✐❝ ✇✐t❤ str♦♥❣ ♣♦❧②♥♦♠✐❛❧ ❞❡❝♦rr❡❧❛t✐♦♥

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✼✮✿ ❋♦r s♠❛❧❧ ❡♥♦✉❣❤✱ s❛t✐s✜❡s ❘❙❲✳ ❘❡♠❛r❦✿ ■❢ ❤❛s ♦s❝✐❧❧❛t✐♥❣ ❝♦rr❡❧❛t✐♦♥s✱ s♦ ❞♦❡s ✳

✸✵✴✺✶

❲✐t❤♦✉t ♣♦s✐t✐✈❡ ❝♦rr❡❧❛t✐♦♥s ✭✇✐t❤♦✉t ❋❑●✮❄

◮ fB : V → R ●❛✉ss✐❛♥ ✜❡❧❞✱

s✐❣♥fB = ❇❡r♥♦✉❧❧✐

◮ f : V → R ●❛✉ss✐❛♥ ✜❡❧❞ ◮ s②♠♠❡tr✐❝ ✇✐t❤ str♦♥❣ ♣♦❧②♥♦♠✐❛❧ ❞❡❝♦rr❡❧❛t✐♦♥

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✼✮✿ ❋♦r ǫ s♠❛❧❧ ❡♥♦✉❣❤✱ fB + ǫf s❛t✐s✜❡s ❘❙❲✳ ❘❡♠❛r❦✿ ■❢ ❤❛s ♦s❝✐❧❧❛t✐♥❣ ❝♦rr❡❧❛t✐♦♥s✱ s♦ ❞♦❡s ✳

✸✵✴✺✶

❲✐t❤♦✉t ♣♦s✐t✐✈❡ ❝♦rr❡❧❛t✐♦♥s ✭✇✐t❤♦✉t ❋❑●✮❄

◮ fB : V → R ●❛✉ss✐❛♥ ✜❡❧❞✱

s✐❣♥fB = ❇❡r♥♦✉❧❧✐

◮ f : V → R ●❛✉ss✐❛♥ ✜❡❧❞ ◮ s②♠♠❡tr✐❝ ✇✐t❤ str♦♥❣ ♣♦❧②♥♦♠✐❛❧ ❞❡❝♦rr❡❧❛t✐♦♥

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✼✮✿ ❋♦r ǫ s♠❛❧❧ ❡♥♦✉❣❤✱ fB + ǫf s❛t✐s✜❡s ❘❙❲✳ ❘❡♠❛r❦✿ ■❢ f ❤❛s ♦s❝✐❧❧❛t✐♥❣ ❝♦rr❡❧❛t✐♦♥s✱ s♦ ❞♦❡s fB + ǫf✳

✸✵✴✺✶

❆ s♠♦♦t❤❡❞ r❛♥❞♦♠ ✇❛✈❡ ✭❙❘❲✮ ♠♦❞❡❧

eRW (x, y) =

• R2 δ1(ξ)eix−y,ξdξ.

■❢ ✐s s♠♦♦t❤ ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt✱ ❞❡❝♦rr❡❧❛t❡s str♦♥❣❧②✳ ■❢ ✐s ❝❧♦s❡ t♦ ✱ t❤❡♥ ♦s❝✐❧❧❛t❡s✳

✸✶✴✺✶

❆ s♠♦♦t❤❡❞ r❛♥❞♦♠ ✇❛✈❡ ✭❙❘❲✮ ♠♦❞❡❧

eRW (x, y) =

• R2 δ1(ξ)eix−y,ξdξ.

eSRW (x, y) =

• ξ∈R2 χ(ξ)eix−y,ξdξ.

◮ ■❢ χ ✐s s♠♦♦t❤ ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt✱ eSRW ❞❡❝♦rr❡❧❛t❡s

str♦♥❣❧②✳

◮ ■❢ χ ✐s ❝❧♦s❡ t♦ δ1✱ t❤❡♥ eSRW ♦s❝✐❧❧❛t❡s✳

✸✶✴✺✶

❈♦r♦❧❧❛r②✿ ❖♥ ❛ ✜①❡❞ V✱ fB + ǫfSRW s❛t✐s✜❡s ❘❙❲ ❢♦r ǫ s♠❛❧❧ ❡♥♦✉❣❤✳

✸✷✴✺✶

❙❘❲ ❛♥❞ ❇❋

✸✸✴✺✶

❚♦② ♠♦❞❡❧

❉❡✜♥✐t✐♦♥✿ g : V → R ❤❛s ✜♥✐t❡ r❛♥❣❡ ℓ ✐❢ x − y > ℓ ⇒ eg(x, y) = 0.

✸✹✴✺✶

❚♦② ♠♦❞❡❧

❉❡✜♥✐t✐♦♥✿ g : V → R ❤❛s ✜♥✐t❡ r❛♥❣❡ ℓ ✐❢ x − y > ℓ ⇒ eg(x, y) = 0.

✸✹✴✺✶

❲✐t❤ ✜♥✐t❡ r❛♥❣❡ ℓ

Pr♦❜ = 1/2.

✸✺✴✺✶

Pr♦❜ = 1/2.

✸✻✴✺✶

▼♦st r✐❣❤t ✈❡rt✐❝❛❧ ❝r♦ss✐♥❣ ✰ ❙②♠♠❡tr✐③❛t✐♦♥

✸✼✴✺✶

❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ③♦♥❡

✸✽✴✺✶

Pr♦❜ ≥ 1/2.

✸✾✴✺✶

Pr♦❜ ≥ 1/2.

✹✵✴✺✶

Pr♦❜ ≥ 1/4.

✹✶✴✺✶

Pr♦❜ ≥ 1/8.

✹✷✴✺✶

Pr♦❜ ≥ 1/8.

✹✸✴✺✶

✹✹✴✺✶

■❢ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❜r✐❞❣❡✳✳✳

◆❡❣❛t✐✈❡ ❛r♠ ❜❡t✇❡❡♥ ℓ ❛♥❞ n✳ ❋♦r ❇❡r♥♦✉❧❧✐✱ Pr♦❜

✹✺✴✺✶

■❢ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❜r✐❞❣❡✳✳✳

◆❡❣❛t✐✈❡ ❛r♠ ❜❡t✇❡❡♥ ℓ ❛♥❞ n✳ ❋♦r ❇❡r♥♦✉❧❧✐✱ Pr♦❜ ≤ ℓ n α>0

✹✺✴✺✶

❈❤♦♦s❡ N s✉❝❤ t❤❛t ❢♦r ❇❡r♥♦✉❧❧✐ Pr♦❜ ≤ 1/32.

✹✻✴✺✶

❚❤❡♥ t❤❡r❡ ❡①✐sts ǫ = ǫ(N) > 0 s✉❝❤ t❤❛t ❢♦r fB + ǫf, Pr♦❜ ≤ 1/16.

✹✼✴✺✶

❋♦r fB + ǫf, Pr♦❜ ≤ 1/16.

✹✽✴✺✶

❋♦r fB + ǫf ❛♥❞ n ≥ N✱ Pr♦❜ ≥ 1/8 − Pr♦❜✭♥♦ ❜r✐❞❣❡✮ ≥ 1/8 − 1/16 = 1/16.

✹✾✴✺✶

❋♦r fB + ǫf, Pr♦❜ ≥ 1/256.

✺✵✴✺✶

❚❤❡♦r❡♠ ✭❇❡✛❛r❛✲● ✷✵✶✼ ❱✳✱ P✐t❡r❜❛r❣ ✶✾✽✷✮ ✿ ▲❡t f : V → R ❜❡ ❛ str♦♥❣❧② ❞❡❝♦rr❡❧❛t✐♥❣ ●❛✉ss✐❛♥ ✜❡❧❞✳ ❚❤❡♥

◮ f ❝❛♥ ❜❡ ❝♦✉♣❧❡❞ ✇✐t❤ g ✇✐t❤

✜♥✐t❡ r❛♥❣❡ √n ≪ n

◮ s✉❝❤ t❤❛t ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t② ♦♥ [0, n]2✱

s✐❣♥ f = s✐❣♥ g.

✺✶✴✺✶