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� Huaiyin Normal University2009.7.30

Terence Tao’s harmonic analysismethod on restricted sumsets

Guo Song

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0. AbstractLet p be a prime, and A, B be finite subsets of Zp. Set A + B = {a + b : a ∈

A, b ∈ B}. Cauchy-Davenport theorem asserts that |A + B| > min{p, |A| +

|B| − 1}. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-

Davenport theorem, by applying a new form of the uncertainty principle on

Fourier transform. Recently Guo song and Sun zhi-wei gave some applications

of Tao’s harmonic analysis method to restricted sumsets. In this article we will

introduce the progress in this field.

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1. Restricted sumsetsLet p be a prime, and A, B be finite subsets of Zp. Set

A + B = {a + b : a ∈ A, b ∈ B} (1)

and

A+B = {a + b : a ∈ A, b ∈ B, a 6= b}. (2)

Cauchy-Davenport theorem asserts that

|A + B| > min{p, |A|+ |B| − 1}. (3)

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A well-known result on restricted sumsets states that

|A+A| > min{p, 2|A| − 3}, (4)

which is conjectured by P.Erdos and H.Heilbronn [6] in 1964 and confirmed

by J.A.Dias da Silva and Y.O.Hamidoune [5] in 1994. In 1995-1996 N.Alon,

M.B.Nathanson and I.Z.Ruzsa[2] proposed a polynomial method in this field

and showed that if |B| > |A| > 0 then

|A+B| > min{p, |A|+ |B| − 2}. (5)

By the powerful polynomial method(cf.[1],[2],[3], [8], [9],[11]), mathemati-

cians gave many interesting results.

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2. Uncertainty principleDifinition 1 Let G be a locally compact Abel group and T be the unit circle

group on complex plane. Let ν : G → T be a continuous group homomor-

phism(such that |ν(x)| = 1 and ν(x + y) = ν(x)ν(y)). The group G of all ν is

called dual group of G.

Difinition 2 The Fourier transform on G is a function FT : L1(G) → C0(G),

f 7→ f such that

f(ν) =

∫ν(−x)f(x)dx, ∀ν ∈ G,

where dx denote the Haar measure on G.

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Let G be the field Zp. For f : Zp → C is any complex-valued function, we may

define its support supp(f) and its Fourier transform f : Zp → C as follow:

supp(f) = {x ∈ Zp : f(x) 6= 0} (6)

and

f(x) =∑a∈Zp

f(a)ep(ax), x ∈ Zp. (7)

where ep(y) = e−2πiy/p for y ∈ Zp.

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Tao obtained the following result in [12] :

Theorem 1 Let p be an odd prime. If f : Zp → C is not identically zero, then

|supp(f)|+ |supp(f)| > p + 1. (8)

Given two non-empty subsets A and B of Zp with |A|+ |B| > p + 1, we can find

a function f : Zp → C with supp(f) = A and supp(f) = B.

Note that the inequality was also discovered independently by Andraas Biro.

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3. Tao’s proof of Cauchy-DavenporttheoremLet m = max{p + 2 − |A| − |B|, 1}, there exist subsets X, Y of Zp such that

|X| = p + 1− |A|, |Y | = p + 1− |B| and |X ∩ Y | = m.

By Theorem 1, there exist functions f, g such that

supp(f) = A, supp(f) = X,

supp(g) = B, supp(g) = Y.

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Let

F (x) = f ∗ g(x) =1

p

∑a∈Zp

f(a)g(x− a).

Observe that

supp(F ) ⊆ supp(f) + supp(g) = A + B

and

supp(F ) ⊆ supp(f) + supp(g) = X ∩ Y,

thus we have

|A + B| ≥ p + 1− |X ∩ Y | = min(p, |A|+ |B| − 1).

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4. Variant of Tao’s method with appli-cation to restricted sumsets

A pair (A, B) of subsets of Zp is said to be m-good if 0 ∈ A and p−m ∈ B

but we don’t have both p−m + t ∈ A and p− t 6∈ B for all t ∈ [0, m− 1].

Let

(A, B)m = {x ∈ Zp : x + t ∈ A and x + m− t ∈ B for some t ∈ [0, m]}.

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Theorem 2 Let A and B be non-empty subsets of Zp with p odd prime, and

C = {a + b : a ∈ A, b ∈ B, a− b 6∈ S} (9)

with S ⊆ Zp and |S| = m. Let A, B be subsets of Zp with |A| > p + 1 − |A|

and |B| > p + 1− |B|. If (A, B) is m-good, then

|C| > p + 1− |(A, B)m|.

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Proof of Theorem 2:

By Theorem 1 there are functions f, g : Zp → C such that supp(f) = A,

supp(f) = A, supp(g) = B and supp(g) = B. Now we define a function

F : Zp → C by

F (x) =∑a∈Zp

f(a)g(x− a)∏d∈S

(ep(x− a)− ep(a− d)). (10)

For each x ∈ supp(F), there exists a ∈ supp(f) with x − a ∈ supp(g) and

d := a− (x− a) 6∈ S, hence x = a + (x− a) ∈ C. Therefore

supp(F) ⊆ C. (11)

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For any x ∈ Z we have

F (x) =∑b∈Zp

F (b)ep(bx) =∑a∈Zp

∑b∈Zp

f(a)g(b− a)ep(bx)P (a, b),

where

P (a, b) =∏d∈S

(ep(b−a)−ep(a−d)) =∑T⊆S

(−1)|T |ep((|S|−|T |)(b−a))ep

(|T |a−

∑d∈T

d).

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Therefore

F (x) =∑T⊆S

(−1)|T |ep

(−

∑d∈T

d) ∑

a∈Zp

f(a)ep(ax+|T |a)×∑b∈Zp

g(b−a)ep((b−a)x+(|S|−|T |)(b−a))

=∑T⊆S

(−1)|T |ep

(−

∑d∈T

d)f(x + |T |)g(x + m− |T |).

By the definition of m-good pair we have

F (p−m) = (−1)mep(−∑d∈S

d)f(0)g(p−m) 6= 0,

so F is not identically zero.

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Suppose that x ∈ supp(F). Then there is a subset T of S with |T | = t such that

x + t ∈ A = supp(f) and x + m− t ∈ B = supp(g), hence x ∈ (A, B)m.

Thus supp(F) ⊆ (A, B)m. With the help of Theorem 1, we have

|C| > |supp(F)| > p + 1− supp(F) > p + 1− |(A, B)m|.

This concludes the proof.

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Construct suitable m-good pair (A, B) so that |[A, B]m| is small and hence |C|

is large.

Example 1 Let

A = {0, 1, . . . , |A| − 1}

and

B = {p− |S| − |B|+ 1, p− |S| − |B|+ 2, . . . , p− |S|},

then (A, B) is m-good.

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Hence,

Theorem 3 Let A and B be non-empty subsets of Zp with p odd prime, and

C = {a + b : a ∈ A, b ∈ B, a− b 6∈ S} (12)

with S ⊆ Zp. Then we have

|C| > min{p, |A|+ |B| − 2|S| − 1}. (13)

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Example 2 Suppose that m+1 6 2|A| 6 |A|+|B| 6 p−m. Set k = p+1−|A|,

l = p + 1 − |B| and n = k − l. Suppose that [[2 - m]] 6 n 6 bm2 c. Let

A = {2i : i = 0, 1, . . . , k − 1} and

B = {x : x ∈ [1, 2k−1−p]}∪{x : x ≡ p−m(mod 2) & x ∈ [2k−p, p+1+2l−2k]}

Then (A, B) is m-good with |(A, B)m| 6 2k + m− p.

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With the help of some m-good pairs, we can obtain that

Theorem 4 Let A and B be non-empty subsets of Zp with p odd prime, and

C = {a + b : a ∈ A, b ∈ B, a− b 6∈ S} (14)

with S ⊆ Zp. Then we have

|C| > min{p, |A|+ |B| − |S| − r}, (15)

where r = 2 if |A| = |B| and |S| ≡ 1(mod 2), r = 1 + min{b |S|2 c, ||A| − |B||}

otherwise.

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In [7] S. Guo and Z.Sun conjecture that min{b |S|2 c, ||A| − |B||} can reduced,

hence r = 2 if |A| = |B| and |S| ≡ 1(mod 2), r = 1 otherwise.

When |S| is even, the conjecture was proposed by Q.Hou and Z.Sun in [8].

In the case ∅ 6= S ⊂ Zp, H. Pan and Z. Sun [11, Corollary 2] obtained the

inequality

|C| > min{p, |A|+ |B| − |S| − 2} (16)

via the polynomial method.

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References

[1] N.Alon, Combinatorial Nullstellensatz, Combin. Prob. Comput., 8, (1999),

7-29.

[2] N.Alon, M.B.Nathanson and I.Z.Ruzsa, The polynomial method and re-

stricted sums of congruence classes, J. Number Theory, 56, (1996), 404-

417.

[3] H.Q.Cao and Z.W.Sun, On sums of distinct representatives, Acta Arith.,

87, (1998), 159-169.

[4] H.Davenport, On the addition of residue classes, J. London Math. Soc., 10,

(1935), 30-32.

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References

[5] J.A.Dias da Silva and Y.O.Hamidoune, Cyclic spaces for Grassmann deriv-

atives and additive theory, Bull. London Math. Soc., 26, (1994), 140-146.

[6] P.Erdos and H.Heilbronn, On the addition of residue classes modulo p,

Acta Arith., 9, (1964), 149-159.

[7] S. Guo and Z.W.Sun, A variant of Tao’s method with application to re-

stricted sumsets , J.Number Theory, 129(2009); 434-438.

[8] Q.H.Hou and Z.W.Sun, Restricted sums in a field, Acta Arith., 102, (2002),

239-249.

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References

[9] J.X.Liu and Z.W.Sun, Sums of subsets with polynomial restrictions, J.

Number Theory, 97, (2002), 301-304.

[10] M.B.Nathanson, Additive Number Theory: Inverse Problems and the

Geometry of Sumsets (Graduated texts in mathematics; 165), Springer,

New York, 1996.

[11] H.Pan and Z.W.Sun, A lower bound for |{a + b : a ∈ A, b ∈ B, P (a, b) 6=

0}|, J. Combin. Theory Ser. A, 100, (2002), 387-393.

[12] Terence Tao, An uncertainty principle for cyclic groups of prime order,

Math. Res. Lett. , 12, (2005), 121-127.

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