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❆ ❙✉❜❧✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞ ♦♥ ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠

❆❧❡①❛♥❞❡r ❘✳ ❇❧♦❝❦ ❖❝t♦❜❡r ✷✱ ✷✵✶✼

✶ ✴ ✸✶

✶ ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠ ✷ ❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✸ Pr♦♦❢ ♦❢ ❖✉r ❘❡s✉❧t ✹ ❈♦♥♥❡❝t✐♦♥ t♦ t❤❡ ❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠

✷ ✴ ✸✶

❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠

■♥ ♦✉r ❈❘❨P❚❖ ✷✵✶✼ P❛♣❡r ❬❇▼◆✶✼❪✱ ✇❡ ✐♥tr♦❞✉❝❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠

❖✉r Pr♦❜❧❡♠

❋✐♥❞ t✇♦ ♦r❞❡r❡❞ s❡ts S = {s1, . . . , sm} ❛♥❞ T = {t1, . . . , tm} s✉❝❤ t❤❛t si ❛♥❞ ti ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ❢♦r ❡✈❡r② i si + tj < n ❢♦r ❡✈❡r② i, j si + tj = sk + tk ❢♦r ❡✈❡r② i, j, k t❤❛t ❛r❡ ♥♦t ❛❧❧ ✐❞❡♥t✐❝❛❧ m ✐s ♠❛①✐♠✐③❡❞ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ✉♣♣❡r ❜♦✉♥❞✐♥❣ m

◗✉❡st✐♦♥

❋♦r ❧❛r❣❡ ❡♥♦✉❣❤ n✱ ✐s m = O(n) ♣♦ss✐❜❧❡❄ m = O(n) ✐s t❤❡ ❜❡st ❜♦✉♥❞ ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r

✸ ✴ ✸✶

❖✉r ❘❡s✉❧ts

❲❡ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t

❚❤❡♦r❡♠ ✭❖✉r Pr♦❜❧❡♠ ✐s ❙✉❜❧✐♥❡❛r✮

❋♦r ♦✉r ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦❜❧❡♠✱ m = O

• n

log∗ n

• ✳

❍❡r❡ log∗ ✐s ❞❡✜♥❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❜❛s❡ ✷ ❛s log∗ n =

• n 1

1 + log∗(log n) n > 1 ❚♦ ❣❛✐♥ ✐♥s✐❣❤t ♦♥ ❤♦✇ t♦ ♣r♦✈❡ t❤✐s r❡s✉❧t✱ ✇❡ ❡①❛♠✐♥❡ s✐♠✐❧❛r ♣r♦❜❧❡♠s ❛♥❞ t❤❡ t❡❝❤♥✐q✉❡s ✉s❡❞ t♦ ♣r♦✈❡ t❤❡♠

✹ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥✿ ✸✲❋r❡❡ ❙❡ts

❖✉r ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦❜❧❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❦♥♦✇ ✸✲❢r❡❡ s❡t ♣r♦❜❧❡♠

✸✲❋r❡❡ ❙❡t Pr♦❜❧❡♠

❋✐♥❞ A ⊆ {1, . . . , n} s✉❝❤ t❤❛t A ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② ✸✲t❡r♠ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ |A| ✐s ♠❛①✐♠✐③❡❞ ❋♦r ♦✉r ♣r♦❜❧❡♠✱ s❡tt✐♥❣ S = T ②✐❡❧❞s t❤❡ ✸✲❋r❡❡ ❙❡t Pr♦❜❧❡♠ ❯♣♣❡r ❜♦✉♥❞s ❢♦r t❤❡ ✸✲❋r❡❡ ❙❡t ♣r♦❜❧❡♠ ♦r✐❣✐♥❛t❡❞ ✇✐t❤ ❘♦t❤ ❬❘♦t✺✸❪

❘♦t❤✬s ❚❤❡♦r❡♠ s❤♦✇s t❤❛t |A| = o(n)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❘♦t❤ ✉s❡s ❋♦✉r✐❡r ❛♥❛❧②s✐s t♦ s❤♦✇ |A| = O(n/(log log n)) ❇❡st ❦♥♦✇♥ ✉♣♣❡r ❜♦✉♥❞✿ O(n(log log n)4/ log n) ❛♥❞ ✇❛s s❤♦✇♥ ❜② ❇❧♦♦♠ ❬❇❧♦✶✻❪✱ ✉s✐♥❣ ❤❡❛✈② ♠❛❝❤✐♥❡r② ❢r♦♠ ❛❞❞✐t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❝s

✺ ✴ ✸✶

Pr♦♦❢ ♦❢ ❛ ❙✉❜❧✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞

❖✉r ❣♦❛❧ ✐s t♦ s❤♦✇ s✉❜❧✐♥❡❛r✐t② ♦❢ ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠

❚❤❡ ♠❛❝❤✐♥❡r② ✐♥ ♠❛♥② ♦❢ t❤❡ ♣r♦♦❢s ❢♦r ✸✲❋r❡❡ ❙❡ts ❞♦ ♥♦t ❡❛s✐❧② ❣❡♥❡r❛❧✐③❡ t♦ ❛♥❛❧②s✐s ♦✈❡r ✷ s❡ts

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ♣r♦♦❢ ♦❢ s✉❜❧✐♥❡❛r✐t② ♦❢ ✸✲❋r❡❡ ❙❡ts t❤❛t ❝❛♥ ❣❡♥❡r❛❧✐③❡ t♦ ♦✉r ♣r♦❜❧❡♠❄

❆♥s✇❡r

❨❡s✦ ❲❡ ❝❛♥ ✉s❡ ❛ ●r❛♣❤ ❚❤❡♦r❡t✐❝ r❡❞✉❝t✐♦♥ ❛♥❞ ✉s❡ ❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

✻ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

❲❡✬❧❧ ❝❛r❡❢✉❧❧② ❞❡✜♥❡ ❛❧❧ t❤❡ ♥❡❝❡ss❛r② ❝♦♠♣♦♥❡♥ts ♥❡❡❞❡❞ ❢♦r ❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛✳ ▲❡t G = (V, E) ❜❡ ❛ s✐♠♣❧❡✱ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ♦♥ n ✈❡rt✐❝❡s ■❢ X ⊆ V ❛♥❞ Y ⊆ V ✱ ✇❡ ❞❡✜♥❡ e(X, Y ) ❛s t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ❜❡t✇❡❡♥ X ❛♥❞ Y

❉❡✜♥✐t✐♦♥ ✭❊❞❣❡ ❉❡♥s✐t②✮

❋♦r ❞✐s❥♦✐♥t X, Y ⊆ V ✱ ✇❡ ❞❡✜♥❡ t❤❡ ❡❞❣❡ ❞❡♥s✐t② ❛s d(X, Y ) = e(X, Y ) |X| · |Y |

✼ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

◆❡①t ✇❡ ❤❛✈❡ t❤❡ ❘❡❣✉❧❛r✐t② ❈♦♥❞✐t✐♦♥

❉❡✜♥✐t✐♦♥ ✭ε✲❘❡❣✉❧❛r P❛✐rs✮

▲❡t ε > 0✳ ●✐✈❡♥ ❞✐s❥♦✐♥t X, Y ⊆ V ✱ t❤❡ ♣❛✐r (X, Y ) ✐s ε✲r❡❣✉❧❛r ✐❢ ❢♦r ❡✈❡r② A ⊆ X ❛♥❞ B ⊆ Y s✉❝❤ t❤❛t |A| > ε|X| ❛♥❞ |B| > ε|Y | ✇❡ ❤❛✈❡ |d(A, B) − d(X, Y )| < ε

✽ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

❚❤❡♦r❡♠ ✭❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❬❙③❡✼✺❜✱ ❑❙❙❙✵✷❪✮

∀ε > 0

• ∃M = M(ε) ∈ Z s✉❝❤ t❤❛t
• ∀ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ G = (V, E) ♦♥ n ✈❡rt✐❝❡s
• ∃ ❛ ♣❛rt✐t✐♦♥ ♦❢ V ✐♥t♦ k ❝❧❛ss❡s V1, V2, . . . , Vk s✉❝❤ t❤❛t
• k M
• |Vi| ⌈εn⌉ ❢♦r ❡✈❡r② i
• ||Vi| − |Vj|| 1 ❢♦r ❛❧❧ i, j ✭❡q✉✐♣❛rt✐t✐♦♥✮
• (Vi, Vj) ✐s ε✲r❡❣✉❧❛r ✐♥ G ❢♦r ❛❧❧ ❡①❝❡♣t ❛t ♠♦st εk2

♣❛✐rs (i, j)

✾ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

❚❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✐s ❛ ✈❡r② ♣♦✇❡r❢✉❧ t♦♦❧ ✐♥ ❊①tr❡♠❛❧ ●r❛♣❤ ❚❤❡♦r②

■♥t✉✐t✐✈❡ ❙t❛t❡♠❡♥t

❆♥② ❧❛r❣❡ ❛♥❞ ❞❡♥s❡ ❡♥♦✉❣❤ ❣r❛♣❤ ✏❧♦♦❦s ❧✐❦❡✑ ❛ ✉♥✐♦♥ ♦❢ ❛ s♠❛❧❧ ❛♠♦✉♥t ♦❢ r❛♥❞♦♠✲❧♦♦❦✐♥❣ ❜✐♣❛rt✐t❡ ❣r❛♣❤s✳ ▲❛r❣❡ ❡♥♦✉❣❤ ❣r❛♣❤s ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ✉♥✐❢♦r♠ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ❋♦r t❤✐s r❡❛s♦♥✱ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✐s ❛❧s♦ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❯♥✐❢♦r♠✐t② ▲❡♠♠❛

◆♦t❡

• ♦✇❡rs ❬●♦✇✾✼❪ s❤♦✇❡❞ t❤❛t t❤❡ ❜♦✉♥❞ ♦❜t❛✐♥❡❞ ❢♦r M(ε) ✐s ❛ t♦✇❡r ♦❢

✷✬s ♦❢ ❤❡✐❣❤t ♣r♦♣♦rt✐♦♥❛❧ t♦ ε−5✳ ❍❡ s❤♦✇❡❞ t❤✐s ✐s ❛♥ ✐♥❤❡r❡♥t ❢❡❛t✉r❡ ♦❢ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

✶✵ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛

❙③❡♠❡ré❞✐ ❝r❡❛t❡❞ t❤✐s ❧❡♠♠❛ t♦ ♣r♦✈❡ ❤✐s ❢❛♠♦✉s r❡s✉❧t

❚❤❡♦r❡♠ ✭❙③❡♠❡ré❞✐✬s ❚❤❡♦r❡♠ ❬❙③❡✼✺❛❪✮

∀k 3 ❛♥❞ δ > 0

• ∃n0 ∈ Z s✉❝❤ t❤❛t
• ∀N n0
• ■❢ A ⊆ {1, 2, . . . , N} ❛♥❞ |A| δN
• t❤❡♥ A ❝♦♥t❛✐♥s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ k

❋♦r k = 3✱ ✇❡ ❤❛✈❡ ❡①❛❝t❧② ❘♦t❤✬s ❚❤❡♦r❡♠ ❢♦r ✸✲❋r❡❡ ❙❡ts

❲❡ ❢♦❧❧♦✇ t❤❡ ♣r♦♦❢ ♦❢ ❘♦t❤✬s ❚❤❡♦r❡♠ ✉s✐♥❣ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ t♦ ♣r♦✈❡ ♦✉r r❡s✉❧t

✶✶ ✴ ✸✶

• r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛

❆s ❛ ❝♦r♦❧❧❛r② ♦❢ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦✇❡r❢✉❧ r❡s✉❧t

▲❡♠♠❛ ✭●r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛ ❬❋♦①✶✶❪✮

▲❡t G ❜❡ ❛♥ n✲✈❡rt❡① ❣r❛♣❤ ❛♥❞ ❧❡t H ❜❡ ❛ ✜①❡❞ h✲✈❡rt❡① ❣r❛♣❤✳ ■❢ G ❝♦♥t❛✐♥s o(nh) ❝♦♣✐❡s ♦❢ H✱ t❤❡♥ G ❝❛♥ ❜❡ ♠❛❞❡ H✲❢r❡❡ ❜② r❡♠♦✈✐♥❣

• (n2) ❡❞❣❡s✳

❇② H✲❢r❡❡✱ ✇❡ ♠❡❛♥ t❤❛t G ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② ❝♦♣✐❡s ♦❢ H ❛s ❛ s✉❜❣r❛♣❤ ❚❤❡ ●r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛ ❝❛♥ ❜❡ r❡st❛t❡❞ ♠♦r❡ ❢♦r♠❛❧❧② ❛s ❢♦❧❧♦✇s

✶✷ ✴ ✸✶

• r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛

▲❡♠♠❛ ✭●r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛ ❬❋♦①✶✶✱ ▲❡❡❪✮

∀ε > 0 ❛♥❞ ❣r❛♣❤s H ♦♥ h ✈❡rt✐❝❡s

• ∃δ > 0 s✉❝❤ t❤❛t
• ∀G = (V, E) ♦♥ n ✈❡rt✐❝❡s
• ✐❢ G ❝♦♥t❛✐♥s ❛t ♠♦st δnh ❝♦♣✐❡s ♦❢ H
• t❤❡♥ G ❝❛♥ ❜❡ ♠❛❞❡ H✲❢r❡❡ ❜② r❡♠♦✈✐♥❣ εn2 ❡❞❣❡s

❲❡ ✉s❡ t❤✐s ✈❡rs✐♦♥ ♦❢ t❤❡ ❧❡♠♠❛ t♦ ♦❜t❛✐♥ ♦✉r ❜♦✉♥❞ ♦♥ m

✶✸ ✴ ✸✶

Pr♦♦❢ ❖✉t❧✐♥❡

❆ ❤✐❣❤✲❧❡✈❡❧ ♦✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢

✶ ❚r❛♥s❢♦r♠ ♦✉r ♣r♦❜❧❡♠ ✐♥t♦ ❛ 3✲♣❛rt✐t❡ ❣r❛♣❤ G ✷ ❈r❡❛t❡ t❤❡ ❡❞❣❡✲s❡t s✉❝❤ t❤❛t G ✐s ❛ ✉♥✐♦♥ ♦❢ ❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s ✸ ❯s❡ t❤❡ ●r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛ t♦ r❡♠♦✈❡ ❛❧❧ tr✐❛♥❣❧❡ ❢r♦♠ G ✹ ❚❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s r❡♠♦✈❡❞ ❡①❛❝t❧② ②✐❡❧❞s ❛ ❜♦✉♥❞ ♦♥ t❤❡ s✐③❡ ♦❢

♦✉r ♣r♦❜❧❡♠

✶✹ ✴ ✸✶

Pr♦♦❢

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t✇♦ s❡ts S ❛♥❞ T ✇❤✐❝❤ s❛t✐s❢② ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠✿

❖✉r Pr♦❜❧❡♠

❋✐♥❞ t✇♦ ♦r❞❡r❡❞ s❡ts S = {s1, . . . , sm} ❛♥❞ T = (t1, . . . , tm) s✉❝❤ t❤❛t si ❛♥❞ ti ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ❢♦r ❡✈❡r② i si + tj < n ❢♦r ❡✈❡r② i, j si + tj = sk + tk ❢♦r ❡✈❡r② i, j, k t❤❛t ❛r❡ ♥♦t ❛❧❧ ✐❞❡♥t✐❝❛❧ m ✐s ♠❛①✐♠✐③❡❞

◆♦t❛t✐♦♥

❋♦r ✐♥t❡❣❡r n✱ ❧❡t [n] := {1, 2, . . . , n} ❈♦♥str✉❝t ❛ ✸✲♣❛rt✐t❡ ❣r❛♣❤ G ✇✐t❤ ✈❡rt❡① s❡ts X = [n]✱ Y = [2n]✱ ❛♥❞ Z = [3n] ❈♦♥str✉❝t t❤❡ ❡❞❣❡ s❡ts ❛s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts

✶✺ ✴ ✸✶

Pr♦♦❢

X 1 2 . . . n Y 1 2 . . . n . . . 2n Z 1 2 . . . n . . . 2n . . . 3n

E(X, Y ) =

• (x, x + si) : x ∈ [n], i ∈ [m], si ∈ S
• E(Y, Z) =
• (x + si, x + si + ti) : x ∈ [n], i ∈ [m], si ∈ S, ti ∈ T
• E(X, Z) =
• (x, x + si + ti) : x ∈ [n], i ∈ [m], si ∈ S, ti ∈ T
• ✶✻ ✴ ✸✶

Pr♦♦❢

❊❞❣❡ ❙❡t ♦❢ G

E(G) ✐s t❤❡ ✉♥✐♦♥ ♦❢ E(X, Y ) =

• (x, x + si) : x ∈ [n], i ∈ [m], si ∈ S
• E(Y, Z) =
• (x + si, x + si + ti) : x ∈ [n], i ∈ [m], si ∈ S, ti ∈ T
• E(X, Z) =
• (x, x + si + ti) : x ∈ [n], i ∈ [m], si ∈ S, ti ∈ T
• ❙✐♥❝❡ x ∈ [n] ❛♥❞ |S| = |T| = m✱ ✇❡ ❤❛✈❡

|E(X, Y )| =|E(Y, Z)| = |E(X, Z)| = nm = ⇒ |E(G)| = 3mn ◆❡①t✱ ✇❡ ❝❧❛✐♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❜♦✉t t❤❡ ❝♦♥str✉❝t❡❞ ❣r❛♣❤ G✿

✶✼ ✴ ✸✶

Pr♦♦❢

❈❧❛✐♠

G ✐s ❛ ✉♥✐♦♥ ♦❢ ❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s✳

◆♦t❡

❆ tr✐❛♥❣❧❡ ✐s ❞❡✜♥❡❞ ❜② t❤❡ t✉♣❧❡ ♦❢ ✈❡rt✐❝❡s (x, y, z) s✉❝❤ t❤❛t

• (x, y), (y, z), (x, z)
• ❛r❡ ❡❞❣❡s ✐♥ G

❋♦r ❛❧❧ x ∈ [n]✱ i ∈ [m]✱ si ∈ S✱ ❛♥❞ ti ∈ T✱ t❤❡ t✉♣❧❡ (x, x + si, x + si + ti) ✐s ❛ tr✐❛♥❣❧❡ ✐♥ G✳ ❚❤✐s ❢♦❧❧♦✇s ❜② ❝♦♥str✉❝t✐♦♥ ♦❢ E(G) s✐♥❝❡

E(X, Y ) =

• (x, x + si) : x ∈ [n], i ∈ [m], si ∈ S
• E(Y, Z) =
• (x + si, x + si + ti) : x ∈ [n], i ∈ [m], si ∈ S, ti ∈ T
• E(X, Z) =
• (x, x + si + ti) : x ∈ [n], i ∈ [m], si ∈ S, ti ∈ T
• ❲❡ ♣r♦❝❡❡❞ ❜② ❝♦♥tr❛❞✐❝t✐♦♥

✶✽ ✴ ✸✶

Pr♦♦❢

❙✉♣♣♦s❡ t❤❡r❡ ❡①✐st t✇♦ ♥♦♥✲❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s (x, x + si, x + si + ti) ❛♥❞ (y, y + sj, y + sj + tj) ✐♥ G ▲❡t (x, x + si) = (y, y + sj) ❜❡ t❤❡ s❤❛r❡❞ ❡❞❣❡

❚❤✐s ✐♠♣❧✐❡s x = y ❛♥❞ si = sj ❙✐♥❝❡ S ✐s ❛ s♦❧✉t✐♦♥ t♦ ♦✉r ♣r♦❜❧❡♠✱ si = sj ✐❢ ❛♥❞ ♦♥❧② ✐❢ i = j✳ ❙✐♥❝❡ i = j✱ ✇❡ ❤❛✈❡ t❤❛t ti = tj ❛❧s♦❀ s♦ t❤❡ t✇♦ tr✐❛♥❣❧❡s ❛r❡ t❤❡ s❛♠❡ ✭❝♦♥tr❛❞✐❝t✐♦♥✮

✶✾ ✴ ✸✶

Pr♦♦❢

❙✉♣♣♦s❡ t❤❡r❡ ❡①✐st t✇♦ ♥♦♥✲❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s (x, x + si, x + si + ti) ❛♥❞ (y, y + sj, y + sj + tj) ✐♥ G ▲❡t (x + si, x + si + ti) = (y + sj, y + sj + tj) ❜❡ t❤❡ s❤❛r❡❞ ❡❞❣❡

❚❤✐s ✐♠♣❧✐❡s t❤❛t x + si = y + sj ✭✶✮ x + si + ti = y + sj + tj ✭✷✮ ❙✉❜st✐t✉t✐♥❣ ❡q✉❛t✐♦♥ ✭✶✮ ✐♥t♦ ✭✷✮ ✇❡ ❤❛✈❡ y + sj + ti = y + sj + tj ❚❤✐s ✐♠♣❧✐❡s ti = tj✱ ✇❤✐❝❤ ❛❣❛✐♥ ✐♠♣❧✐❡s i = j ❙♦ si = sj ❛♥❞ ❜② ❡q✉❛t✐♦♥ ✭✶✮✱ x = y ❆❣❛✐♥✱ t❤❡s❡ tr✐❛♥❣❧❡s ❛r❡ t❤❡ s❛♠❡ ✭❝♦♥tr❛❞✐❝t✐♦♥✮

✷✵ ✴ ✸✶

Pr♦♦❢

❙✉♣♣♦s❡ t❤❡r❡ ❡①✐st t✇♦ ♥♦♥✲❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s (x, x + si, x + si + ti) ❛♥❞ (y, y + sj, y + sj + tj) ✐♥ G ▲❡t (x, x + si + ti) = (y, y + sj + tj) ❜❡ t❤❡ s❤❛r❡❞ ❡❞❣❡

❚❤✐s ✐♠♣❧✐❡s x = y ❛♥❞ si + ti = sj + tj ■❢ i = j✱ t❤❡♥ (si, ti) ❛♥❞ (sj, tj) ❛r❡ ❛ ✇✐t♥❡ss t♦ S ❛♥❞ T ♥♦t s❛t✐s❢②✐♥❣ ♦✉r ♣r♦❜❧❡♠ ✭❝♦♥tr❛❞✐❝ts ♦✉r ❛ss✉♠♣t✐♦♥ ❛❜♦✉t S ❛♥❞ T✮ ■t ♠✉st ❜❡ t❤❡ ❝❛s❡ i = j✱ ❛♥❞ ❛❣❛✐♥ t❤❡s❡ t✇♦ tr✐❛♥❣❧❡s ❛r❡ t❤❡ s❛♠❡ ✭❝♦♥tr❛❞✐❝t✐♦♥✮

❚❤❡r❡❢♦r❡✱ ❛❧❧ tr✐❛♥❣❧❡s ❛r❡ ❡❞❣❡✲❞✐s❥♦✐♥t✦

✷✶ ✴ ✸✶

Pr♦♦❢

❙✐♥❝❡ G ✐s ❛ ✉♥✐♦♥ ♦❢ ❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s✱ ✇❡ ❤❛✈❡ t❤❡ ♥✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s ✐♥ G ✐s ❡①❛❝t❧② nm ❙✐♥❝❡ S, T ⊆ [n]✱ ✇❡ ❤❛✈❡ t❤❛t nm n2 = o(n3) ❋♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ n ✇❡ ❦♥♦✇ n2 < δn3✱ ✇❤❡r❡ δ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡

• r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛

❙♦ G ❤❛s ❛t ♠♦st δn3 tr✐❛♥❣❧❡s

✷✷ ✴ ✸✶

Pr♦♦❢

❚❤✉s ❜② t❤❡ ●r❛♣❤ ❘❡♠♦✈❛❧ ▲❡♠♠❛✱ ✇❡ ❝❛♥ r❡♠♦✈❡ εn2 ❡❞❣❡s t♦ ♠❛❦❡ G tr✐❛♥❣❧❡✲❢r❡❡ ❙✐♥❝❡ t❤❡r❡ ❛r❡ nm ❡❞❣❡✲❞✐s❥♦✐♥t tr✐❛♥❣❧❡s ✐♥ G✱ ✐t ✐s s✉✣❝✐❡♥t t♦ r❡♠♦✈❡ ♦♥❡ ❡❞❣❡ ❢r♦♠ ❡❛❝❤ tr✐❛♥❣❧❡ t♦ ❞❡str♦② ❛❧❧ tr✐❛♥❣❧❡s ✐♥ G

❙♦ ✇❡ ❝❛♥ r❡♠♦✈❡ nm ❡❞❣❡s t♦ ❞❡str♦② ❛❧❧ tr✐❛♥❣❧❡s ❚❤✐s ✐♠♣❧✐❡s nm = εn2✱ ❛♥❞ t❤✉s m = εn

✷✸ ✴ ✸✶

Pr♦♦❢

❆s ♥♦t❡❞ ✐♥ ❬❋♦①✶✶✱ ▲❡❡❪✱ δ−1 ✐s ❛ t♦✇❡r ♦❢ t✇♦s ♦❢ ❤❡✐❣❤t ♣r♦♣♦rt✐♦♥❛❧ t♦ ε−5

❈❤♦♦s❡ n ❧❛r❣❡ ❡♥♦✉❣❤ s✉❝❤ t❤❛t log∗ n = ε−5 ❚❤✐s ✐♠♣❧✐❡s δ−1 = 22. . .2 O(ε−5) = 22. . .2 O(log∗ n) = O(n) ❚❤✐s ✐♠♣❧✐❡s t❤❛t εδ−1 = O(εn) = O

• n

log∗ n

• < n

✷✹ ✴ ✸✶

Pr♦♦❢

■♥ ♣❛rt✐❝✉❧❛r✱ εn = O

• n

log∗ n

• ❚❤✐s ✐♠♣❧✐❡s

m = εn = O

• n

log∗ n

• ✷✺ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥ t♦ ❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠

❲❡ s❛✇ ❜❡❢♦r❡ t❤❛t ♦✉r ♣r♦❜❧❡♠ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ✈❡rs✐♦♥ ♦❢ t❤❡ ✸✲❋r❡❡ ❙❡t ♣r♦❜❧❡♠

◗✉❡st✐♦♥

■s ♦✉r ♣r♦❜❧❡♠ r❡❧❛t❡❞ t♦ ♦t❤❡r ♣r♦❜❧❡♠s ✐♥ ❛❞❞✐t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❝s❄

❆♥s✇❡r

❨❡s✱ ❚❤❡ ❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠✦

❉❡✜♥✐t✐♦♥ ✭❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t ✭❚❈❙❋✮ ❬❇❈❈+✶✼❪✮

❋♦r ❛♥② ❛❜❡❧✐❛♥ ❣r♦✉♣ G✱ ❛ tr✐✲❝♦❧♦r❡❞ s✉♠✲❢r❡❡ s❡t ✐♥ G ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ tr✐♣❧❡s {(ai, bj, ck)} ✐♥ G s✉❝❤ t❤❛t ai + bj + ck = 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ i = j = k✳ ❚❤❡ ❚❈❙❋ ❙❡t ♣r♦❜❧❡♠ ❛s❦s ✇❤❛t ✐s t❤❡ ❧❛r❣❡st ❚❈❙❋ ❙❡t ✇❤✐❝❤ ❝❛♥ ♦❝❝✉r ✐♥ G

Pr♦❜❧❡♠ ✐s ❛❧s♦ ❦♥♦✇s ❛s t❤❡ ❧❛r❣❡st ❝❛♣ s❡t ♣r♦❜❧❡♠

✷✻ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥ t♦ ❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠

◆♦t✐❝❡ ❢♦r G = Z✱ t❤❡ s❡t {(x, x, −2x)} ❢♦r x ∈ Z ✐s ❡①❛❝t❧② t❤❡ ✸✲❋r❡❡ ❙❡t ♣r♦❜❧❡♠ ■♥ ❢❛❝t ❢♦r G = Z✱ t❤❡ ❚❈❙❋ ❙❡t {(si, tj, −(sk + tk))} ✐s ✐❞❡♥t✐❝❛❧ t♦ ♦✉r ♣r♦❜❧❡♠✦

✷✼ ✴ ✸✶

❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠

❇❡st✲❦♥♦✇♥ ✉♣♣❡r ❜♦✉♥❞ r❡s✉❧ts ❛❜♦✉t t❤❡ ❚❈❙❋ ✇❡r❡ ♣r♦✈❡♥ ♦♥❧② r❡❝❡♥t❧② ✭▼❛② ✷✵✶✻✮

❚❤❡♦r❡♠ ✭❬❇❈❈+✶✼❪✱ ❚❤❡♦r❡♠ ✹✳✶✹✮

■❢ q ✐s ❛ ♣r✐♠❡ ♣♦✇❡r ❛♥❞ Cq ✐s t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ ♦❢ ♦r❞❡r q✱ t❤❡♥ s✉♠✲❢r❡❡ s❡ts ✐♥ Cn

q ❤❛✈❡ s✐③❡ ❛t ♠♦st 3θn✱ ✇❤❡r❡

θ = min

ρ>0 (1 + ρ + · · · + ρq−1)ρ−(q−1)/3

✷✽ ✴ ✸✶

❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠

▼♦r❡ r❡❝❡♥t❧② ✭❏✉♥❡ ✷✵✶✻✮✱ t❤✐s r❡s✉❧t ✇❛s s❤♦✇♥ t♦ ❜❡ t✐❣❤t ✇✐t❤✐♥ ❛ s✉❜❡①♣♦♥❡♥t✐❛❧ ❢❛❝t♦r

❚❤❡♦r❡♠ ✭❬❑❙❙✶✻❪✱ ❚❤❡♦r❡♠ ✷✮

❋✐① ❛♥ ✐♥t❡❣❡r q 2 ❛♥❞ ❞❡✜♥❡ θ ❛s ❛❜♦✈❡✳ ❋♦r n s✉✣❝✐❡♥t❧② ❧❛r❣❡✱ t❤❡r❡ ❛r❡ s✉♠✲❢r❡❡ s❡ts ✐♥ Cn

q ✇✐t❤ s✐③❡ ❛t ❧❡❛st

θn exp

• −2
• 2(log 2)(log θ)n − Oq(log n)
• ✇❤❡r❡ Oq(·) r❡❢❡rs t♦ ❜♦✉♥❞s ❛s n → ∞ t❤r♦✉❣❤ ✐♥t❡❣❡rs ❞✐✈✐s✐❜❧❡

❜② 3 ❛♥❞ q ✜①❡❞

✷✾ ✴ ✸✶

❋✐♥❛❧ ❘❡♠❛r❦s

❯♣♣❡r ❜♦✉♥❞s ❜❡❛t✐♥❣ O(n/ log∗ n) ❢♦r G = Z ✐♥ t❤❡ ❚❈❙❋ ❙❡t Pr♦❜❧❡♠ ❤❛✈❡ ♥♦t ❜❡❡♥ s❤♦✇♥ ②❡t ❚❤✐s ✐s ❛♥ ❛❝t✐✈❡ ♣r♦❜❧❡♠ ✐♥ ❛❞❞✐t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❝s ✇✐t❤ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s

▼❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❬❑❙❙✶✻❪ Pr♦♣❡rt② t❡st✐♥❣ ❬❇❳✶✺❪ ❘❡♠♦✈❛❧ ❧❡♠♠❛s ✐♥ ❛❞❞✐t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❝s ❬●r❡✵✺❪

✸✵ ✴ ✸✶

❚❤❛♥❦ ❨♦✉✦

❆♥② q✉❡st✐♦♥s❄

✸✶ ✴ ✸✶

❬❇❈❈+✶✼❪ ❏♦♥❛❤ ❇❧❛s✐❛❦✱ ❚❤♦♠❛s ❈❤✉r❝❤✱ ❍❡♥r② ❈♦❤♥✱ ❏♦s❤✉❛ ❆✳

• r♦❝❤♦✇✱ ❊r✐❝ ◆❛s❧✉♥❞✱ ❲✐❧❧✐❛♠ ❋✳ ❙✇❛✐♥✱ ❛♥❞ ❈❤r✐s

❯♠❛♥s✳ ❖♥ ❝❛♣ s❡ts ❛♥❞ t❤❡ ❣r♦✉♣✲t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤ t♦ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❉✐s❝r❡t❡ ❆♥❛❧②s✐s✱ ✷✵✶✼✳ ❬❇❧♦✶✻❪ ❚✳ ❋✳ ❇❧♦♦♠✳ ❆ q✉❛♥t✐t❛t✐✈❡ ✐♠♣r♦✈❡♠❡♥t ❢♦r r♦t❤✬s t❤❡♦r❡♠ ♦♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✾✸✭✸✮✿✻✹✸✕✻✻✸✱ ✷✵✶✻✳ ❬❇▼◆✶✼❪ ❆❧❡①❛♥❞❡r ❘✳ ❇❧♦❝❦✱ ❍❡♠❛♥t❛ ❑✳ ▼❛❥✐✱ ❛♥❞ ❍❛✐ ❍✳ ◆❣✉②❡♥✳ ❙❡❝✉r❡ ❝♦♠♣✉t❛t✐♦♥ ❜❛s❡❞ ♦♥ ❧❡❛❦② ❝♦rr❡❧❛t✐♦♥s✿ ❍✐❣❤ r❡s✐❧✐❡♥❝❡ s❡tt✐♥❣✳ ■♥ ❏♦♥❛t❤❛♥ ❑❛t③ ❛♥❞ ❍♦✈❛✈ ❙❤❛❝❤❛♠✱ ❡❞✐t♦rs✱ ❆❞✈❛♥❝❡s ✐♥ ❈r②♣t♦❧♦❣② ✕ ❈❘❨P❚❖ ✷✵✶✼✱ P❛rt ■■✱ ✈♦❧✉♠❡ ✶✵✹✵✷ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✕✸✷✱ ❙❛♥t❛

✸✶ ✴ ✸✶

❇❛r❜❛r❛✱ ❈❆✱ ❯❙❆✱ ❆✉❣✉st ✷✵✕✷✹✱ ✷✵✶✼✳ ❙♣r✐♥❣❡r✱ ❍❡✐❞❡❧❜❡r❣✱ ●❡r♠❛♥②✳ ❬❇❳✶✺❪ ❆r♥❛❜ ❇❤❛tt❛❝❤❛r②❛ ❛♥❞ ◆✐♥❣ ❳✐❡✳ ▲♦✇❡r ❜♦✉♥❞s ❢♦r t❡st✐♥❣ tr✐❛♥❣❧❡ ❢r❡❡♥❡ss ✐♥ ❜♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❈♦♠♣❧❡①✐t②✱ ✷✹✭✶✮✿✻✺✕✶✵✶✱ ✷✵✶✺✳ ❬❋♦①✶✶❪ ❏❛❝♦❜ ❋♦①✳ ❆ ♥❡✇ ♣r♦♦❢ ♦❢ t❤❡ ❣r❛♣❤ r❡♠♦✈❛❧ ❧❡♠♠❛✳ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✭✷✮✱ ✶✼✹✭✶✮✿✺✻✶✕✺✼✾✱ ✷✵✶✶✳ ❬●♦✇✾✼❪ ❲✳❚✳ ●♦✇❡rs✳ ▲♦✇❡r ❜♦✉♥❞s ♦❢ t♦✇❡r t②♣❡ ❢♦r s③❡♠❡ré❞✐✬s ✉♥✐❢♦r♠✐t② ❧❡♠♠❛✳

• ❡♦♠❡tr✐❝ ✫ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ●❆❋❆✱ ✼✭✷✮✿✸✷✷✕✸✸✼✱ ▼❛②

✶✾✾✼✳ ❬●r❡✵✺❪ ❇❡♥ ●r❡❡♥✳ ❆ s③❡♠❡ré❞✐✲t②♣❡ r❡❣✉❧❛r✐t② ❧❡♠♠❛ ✐♥ ❛❜❡❧✐❛♥ ❣r♦✉♣s✱ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✳

✸✶ ✴ ✸✶

• ❡♦♠❡tr✐❝ ✫ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ✭●❆❋❆✮✱ ✶✺✭✷✮✿✶✾✸✕✷✵✼✱

✷✵✵✺✳ ❬❑❙❙✶✻❪ ❘✳ ❑❧❡✐♥❜❡r❣✱ ❲✳ ❋✳ ❙❛✇✐♥✱ ❛♥❞ ❉✳ ❊✳ ❙♣❡②❡r✳ ❚❤❡ ●r♦✇t❤ ❘❛t❡ ♦❢ ❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡ts✳ ❆r❳✐✈ ❡✲♣r✐♥ts✱ ❏✉♥❡ ✷✵✶✻✳ ❬❑❙❙❙✵✷❪ ❏á♥♦s ❑♦♠❧ós✱ ❆❧✐ ❙❤♦❦♦✉❢❛♥❞❡❤✱ ▼✐❦❧ós ❙✐♠♦♥♦✈✐ts✱ ❛♥❞ ❊♥❞r❡ ❙③❡♠❡ré❞✐✳ ❚❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ●r❛♣❤ ❚❤❡♦r②✱ ♣❛❣❡s ✽✹✕✶✶✷✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✷✳ ❬▲❡❡❪ ❈❤♦♦♥❣❜✉♠ ▲❡❡✳ ▲❡❝t✉r❡ ✸✿ ❘❡❣✉❧❛r✐t② ❧❡♠♠❛✳ ❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴♠❛t❤✳♠✐t✳❡❞✉✴⑦❝❜❴❧❡❡✴✶✽✳✸✶✽✴❧❡❝t✉r❡✸✳♣❞❢ ✭❙♣r✐♥❣ ✷✵✶✺✮✳ ❬❘♦t✺✸❪ ❑❧❛✉s ❋ ❘♦t❤✳ ❖♥ ❝❡rt❛✐♥ s❡ts ♦❢ ✐♥t❡❣❡rs✳

✸✶ ✴ ✸✶

❏♦✉r♥❛❧ ♦❢ t❤❡ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✭✶✮✿✶✵✹✕✶✵✾✱ ✶✾✺✸✳ ❬❙③❡✼✺❛❪ ❊♥❞r❡ ❙③❡♠❡ré❞✐✳ ❖♥ s❡ts ♦❢ ✐♥t❡❣❡rs ❝♦♥t❛✐♥✐♥❣ ❦ ❡❧❡♠❡♥ts ✐♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✳ ❆❝t❛ ❆r✐t❤♠❡t✐❝❛✱ ✷✼✭✶✮✿✶✾✾✕✷✹✺✱ ✶✾✼✺✳ ❬❙③❡✼✺❜❪ ❊♥❞r❡ ❙③❡♠❡ré❞✐✳ ❘❡❣✉❧❛r ♣❛rt✐t✐♦♥s ♦❢ ❣r❛♣❤s✳ ❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱ ❙t❛♥❢♦r❞✱ ❈❆✱ ❯❙❆✱ ✶✾✼✺✳

✸✶ ✴ ✸✶