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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 ? ✺✴✺✶

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

❙q✉❛r❡s ❲✐t❤ ◮ s②♠♠❡tr② ❜❡t✇❡❡♥ ✰ ❛♥❞ ✲ ◮ s②♠♠❡tr② ❜❡t✇❡❡♥ x 1 ❛♥❞ x 2 t❤❡♥ ❜♦t❤ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡q✉❛❧✳✳✳ ✻✴✺✶

Pr♦❜ = 1 / 2 . ✼✴✺✶

❇♦♥❞ ♣❡r❝♦❧❛t✐♦♥ ♦♥ Z 2 ✳ ❚❤❡♦r❡♠ ✭❘✉ss♦✱ ❙❡②♠♦✉r✲❲❡❧s❤ ✶✾✼✽✮ ▲❡t R ⊂ R 2 ✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts c > 0 ✱ lim inf n →∞ Pr♦❜ ( ♣♦s✐t✐✈❡ ❝r♦ss✐♥❣ ♦❢ nR ) > c. ✽✴✺✶

◗✉❡st✐♦♥ ✿ ▲❡t f : R 2 → R ❛ ❜❡ r❛♥❞♦♠ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ❛♥❞ ✜① R ⊂ R 2 ✳ ❉♦❡s ✐t ❡①✐st c > 0 ✱ � � lim inf n →∞ Pr♦❜ { f > 0 } ❝r♦ss❡s nR > c ? ✾✴✺✶

❚✇♦ ✉♥✐✈❡rs❛❧ ♠♦❞❡❧s ❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧ ✭❘❲✮ ✭❘✐❡♠❛♥♥✐❛♥✮ ❚❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ ♠♦❞❡❧ ✭❛❧❣❡❜r❛✐❝✮ ▲❡t f : R 2 → R ❜❡ ◮ ❛ ❝❡♥t❡r❡❞ ●❛✉ss✐❛♥ ✜❡❧❞ ◮ ✇✐t❤ s②♠♠❡tr✐❝ ❝♦✈❛r✐❛♥t ❢✉♥❝t✐♦♥ e ( x, y ) := E ( f ( x ) f ( y )) = k ( � x − y � ) . ✶✵✴✺✶

▲❡t f : R 2 → 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 ♠♦❞❡❧ ❢♦r t❤❡ r❡s❝❛❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s ✭❛♥❞ ♠♦r❡ ✲ ✉♥✐✈❡rs❛❧ ❢r♦♠ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s✮✳ ❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧ ❇❛r♥❡tt✱ ❇♦❣♦♠♦❧♥②✲❙❝❤♠✐❞t ◮ g ( r, θ ) = � ∞ m = −∞ a m J | m | ( r ) e imθ ✶✶✴✺✶

❚❤❡ r❛♥❞♦♠ ✇❛✈❡ ♠♦❞❡❧ ❇❛r♥❡tt✱ ❇♦❣♦♠♦❧♥②✲❙❝❤♠✐❞t ◮ g ( r, θ ) = � ∞ m = −∞ a m J | m | ( r ) e imθ ◮ ❧✐♠✐t ♠♦❞❡❧ ❢♦r t❤❡ r❡s❝❛❧❡❞ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s ◮ ✭❛♥❞ ♠♦r❡ ✲ ✉♥✐✈❡rs❛❧ ❢r♦♠ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞s✮✳ ✶✶✴✺✶

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

◆❛st❛s❡s❝✉ ✲ ✐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❛ 1 x j x i ◮ f ( x 1 , x 2 ) = � ∞ i,j =0 a ij √ i ! j ! 2 ✶✸✴✺✶

❚❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ ♠♦❞❡❧ ◆❛st❛s❡s❝✉ ✲ ❇❡✛❛r❛ 1 x j x i ◮ f ( x 1 , x 2 ) = � ∞ i,j =0 a ij √ i ! j ! 2 ◮ ✐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❣♠❛♥♥✲❋♦❝❦✱ � ℓ � α> 0 Pr♦❜ < n ✶✻✴✺✶

❚❤❡♦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♦❧❧❛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❡✳ ✶✽✴✺✶

✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡ ❘❛♥❞♦♠ ✇❛✈❡s✿ ✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② str♦♥❣ ❞❡♣❡♥❞❡♥❝❡ ❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄ ◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿ e ( x, y ) = exp( −� x − y � 2 ) . ✶✾✴✺✶

❘❛♥❞♦♠ ✇❛✈❡s✿ ✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② str♦♥❣ ❞❡♣❡♥❞❡♥❝❡ ❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄ ◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿ e ( x, y ) = exp( −� x − y � 2 ) . ✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② → ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡ ✶✾✴✺✶

✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② str♦♥❣ ❞❡♣❡♥❞❡♥❝❡ ❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄ ◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿ e ( x, y ) = exp( −� x − y � 2 ) . ✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② → ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡ ◮ ❘❛♥❞♦♠ ✇❛✈❡s✿ e ( x, y ) = J 0 ( � x − y � ) ✶✾✴✺✶

❲❤② ❇❛r❣♠❛♥♥✲❋♦❝❦ ❛♥❞ ♥♦t ❘❛♥❞♦♠ ❲❛✈❡s❄ ◮ ❇❛r❣♠❛♥♥✲❋♦❝❦✿ e ( x, y ) = exp( −� x − y � 2 ) . ✶✳ ♣♦s✐t✐✈❡ ✷✳ ❢❛st ❞❡❝❛② → ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡ ◮ ❘❛♥❞♦♠ ✇❛✈❡s✿ e ( x, y ) = J 0 ( � x − y � ) ✶✳ ♦s❝✐❧❧❛t✐♥❣ ✷✳ s❧♦✇ ❞❡❝❛② → str♦♥❣ ❞❡♣❡♥❞❡♥❝❡ ✶✾✴✺✶

✳✳✳ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❆♥❛❧②t✐❝ ❈♦♥t✐♥✉❛t✐♦♥ P❤❡♥♦♠❡♥♦♥✳ ❙tr♦♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐s ♥♦t ❡♥♦✉❣❤✳✳✳ ✷✵✴✺✶

✳✳✳ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❆♥❛❧②t✐❝ ❈♦♥t✐♥✉❛t✐♦♥ P❤❡♥♦♠❡♥♦♥✳ ❙tr♦♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐s ♥♦t ❡♥♦✉❣❤✳✳✳ ✷✵✴✺✶

❙tr♦♥❣ ❞❡❝♦rr❡❧❛t✐♦♥ ✐s ♥♦t ❡♥♦✉❣❤✳✳✳ ✳✳✳ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❆♥❛❧②t✐❝ ❈♦♥t✐♥✉❛t✐♦♥ P❤❡♥♦♠❡♥♦♥✳ ✷✵✴✺✶