In this paper, we develop Terence Tao's harmonic analysis method and apply it
to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if
$A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge
min{p,|A|+|B|-1}$, where $A+B={a+b: a\in A, b\in B}$. 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. We modify Tao's method
so that it can be used to prove the following extension of the Erdos-Heilbronn
conjecture: If $A,B,S$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then
$|{a+b: a\in A, b\in B, a-b not\in S}|\ge min {p,|A|+|B|-2|S|-1}$

Guo, Song

Sun, Zhi-Wei

English

arXiv

AbstractIn this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well-known Cauchy–Davenport theorem asserts that if ∅≠A, B⊆Z/pZ with p a prime, then |A+B|⩾min{p,|A|+|B|−1}, where A+B={a+b:a∈A,b∈B}. 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. We modify Tao's method so that it can be used to prove the following extension of the Erdős–Heilbronn conjecture: If A,B,S are non-empty subsets of Z/pZ with p a prime, then |{a+b:a∈A,b∈B,a−b∉S}|⩾min{p,|A|+|B|−2|S|−1}

Elsevier - Publisher Connector 

A variant of Tao's method with application to restricted sumsets 

Journal of Number Theory 129 (2009) 434–438Contents lists available at ScienceDirect
Journal of Number Theory
www.elsevier.com/locate/jnt
A variant of Tao’s method with application to restricted
sumsets✩
Song Guo a,∗, Zhi-Wei Sun b
a Department of Mathematics, Huaiyin Teachers College, Huaian 223300, People’s Republic of China
b Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
a r t i c l e i n f o a b s t r a c t
Article history:
Received 20 December 2007
Revised 29 May 2008
Available online 19 August 2008
Communicated by David Goss
MSC:
primary 11B75
secondary 05A05, 11P99, 11T99
Keywords:
Restricted sumsets
Uncertainty principle
Erdo˝s–Heilbronn conjecture
In this paper, we develop Terence Tao’s harmonic analysis method
and apply it to restricted sumsets. The well-known Cauchy–
Davenport theorem asserts that if ∅ = A, B ⊆ Z/pZ with p a prime,
then |A + B|min{p, |A| + |B| − 1}, where A + B = {a + b: a ∈ A,
b ∈ B}. 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. We modify Tao’s
method so that it can be used to prove the following extension of
the Erdo˝s–Heilbronn conjecture: If A, B, S are non-empty subsets
of Z/pZ with p a prime, then |{a + b: a ∈ A, b ∈ B, a − b /∈ S}|
min{p, |A| + |B| − 2|S| − 1}.
© 2008 Elsevier Inc. All rights reserved.
1. Introduction
Let p be a prime, and let A and B be two subsets of the ﬁnite ﬁeld
Zp = Z/pZ = {r¯ = r + pZ: r ∈ Z}.
Set
A + B = {a + b: a ∈ A, b ∈ B} (1)
✩ The ﬁrst author is supported by Program for Excellent Talents in Huaiyin Teachers College. The second author is supported
by the National Natural Science Foundation of People’s Republic of China.
* Corresponding author.
E-mail addresses: guosong77@hytc.edu.cn (S. Guo), zwsun@nju.edu.cn (Z.-W. Sun).
URL: http://math.nju.edu.cn/?zwsun (Z.-W. Sun).0022-314X/$ – see front matter © 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.jnt.2008.05.009
S. Guo, Z.-W. Sun / Journal of Number Theory 129 (2009) 434–438 435and
A  B = {a + b: a ∈ A, b ∈ B, a = b}. (2)
The well-known Cauchy–Davenport theorem (cf. [3] and [9, p. 44]) asserts that
|A + B|min{p, |A| + |B| − 1}. (3)
In 1964 P. Erdo˝s and H. Heilbronn [5] conjectured that
|A  A|min{p,2|A| − 3}; (4)
this was conﬁrmed by J.A. Dias da Silva and Y.O. Hamidoune [4] in 1994. In 1995–1996 N. Alon,
M.B. Nathanson and I.Z. Ruzsa [2] proposed the so-called polynomial method to handle similar prob-
lems. By the powerful polynomial method (cf. [1,2]), many interesting results on restricted sumsets
have been obtained (see, e.g., [7,8,10–12]).
In 2005, Terence Tao [13] developed a harmonic analysis method in this area, applying a new form
of the uncertainty principle on Fourier transform. Let p be a prime. For a complex-valued function
f : Zp → C, we deﬁne its support supp( f ) and its Fourier transform fˆ : Zp → C as follows:
supp( f ) = {x ∈ Zp: f (x) = 0} (5)
and
fˆ (x) =
∑
a∈Zp
f (a)ep(ax) for all x ∈ Zp, (6)
where ep(r¯) = e−2π ir/p for any r ∈ Z.
Here is the main result of the paper [13].
Theorem 1. (See T. Tao [13].) Let p be an odd prime. If f : Zp → C is not identically zero, then
∣∣supp( f )∣∣+ ∣∣supp( fˆ )∣∣ p + 1. (7)
Moreover, given two non-empty subsets A and B of Zp with |A| + |B|  p + 1, we can ﬁnd a function
f :Zp → C such that supp( f ) = A and supp( fˆ ) = B.
Using this theorem Tao [13] gave a new proof of Cauchy–Davenport theorem. Note that the in-
equality (7) was also discovered independently by Andráas Biró (cf. [6,13]). In this article we adapt
the method further and use the reﬁned method to deduce the following result.
Theorem 2. Let A and B be non-empty subsets of Zp with p a prime, and let
C = {a + b: a ∈ A, b ∈ B, a − b /∈ S} (8)
with S ⊆ Zp . Then we have
|C |min{p, |A| + |B| − 2|S| − 1}. (9)
436 S. Guo, Z.-W. Sun / Journal of Number Theory 129 (2009) 434–438Theorem 2 in the case S = ∅ reduces to the Cauchy–Davenport theorem. When A = B and S = {0},
Theorem 2 yields the Erdo˝s–Heilbronn conjecture. In the case p = 2 and ∅ = S ⊂ Zp , Pan and Sun [10,
Corollary 2] obtained the stronger inequality
|C |min{p, |A| + |B| − |S| − 2} (10)
via the polynomial method. The second author (cf. [7]) ever conjectured that 2 in (10) can be replaced
by 1 if |S| is even. We conjecture that when A = B we can also substitute 1 for 2 in (10).
2. Proof of Theorem 2
Without loss of generality, we let |A| |B|. When |A|+ |B| 2|S|+1 or |A| = 1, (9) holds trivially.
Below we suppose that |A| + |B| > 2|S| + 1 and |A| 2.
In the case p = 2, we have A = B = Z2 and C = (A + B) \ S = Z2 \ S , thus
|C | = 2− |S|min{2, |A| + |B| − 2|S| − 1}= min{2,3− 2|S|}.
Below we assume that p is an odd prime. Set k = p − |A| + 1 ∈ [1, p − 1] and l = p − |B| + 1 ∈
[1, p − 1]. Then k + l 2p − 2|S| and l p − |S| since 2|B| |A| + |B| 2|S| + 2. Deﬁne
Aˆ = {0¯, . . . ,k − 1} = {0¯, . . . , p − |A|} (11)
and
Bˆ = {p − |S| − l + 1, . . . , p − |S|}= {|B| − |S|, . . . , p − |S|}. (12)
Clearly, | Aˆ| = p + 1− |A| and |Bˆ| = p + 1− |B|. By Theorem 1 there are functions f , g : Zp → C such
that
supp( f ) = A, supp( fˆ ) = Aˆ, supp(g) = B, supp(gˆ) = Bˆ. (13)
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)
)
. (14)
For each x ∈ supp(F ), there exists a ∈ supp( f ) with x − a ∈ supp(g) and d := a − (x − a) /∈ S , hence
x = a + (x− a) ∈ C . Therefore
supp(F ) ⊆ C . (15)
For any x ∈ Z we have
Fˆ (x) =
∑
b∈Zp
F (b)ep(bx) =
∑
a∈Zp
∑
b∈Zp
f (a)g(b − a)ep(bx)P (a,b),
where
S. Guo, Z.-W. Sun / Journal of Number Theory 129 (2009) 434–438 437P (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
)
.
Therefore
Fˆ (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
)
fˆ
(
x+ |T |)gˆ(x+ |S| − |T |).
For T ⊆ S , if p − |S| + |S| − |T | ∈ supp(gˆ) = Bˆ , then we must have |T | = |S| (i.e., T = S) by the
deﬁnition of Bˆ . It follows that
Fˆ
(
p − |S|)= (−1)|S|ep
(
−
∑
d∈S
d
)
fˆ (0¯)gˆ
(
p − |S|) = 0
since 0¯ ∈ Aˆ = supp( fˆ ) and p − |S| ∈ Bˆ = supp(gˆ). With the helps of (15) and Theorem 1, we get
|C | ∣∣supp(F )∣∣ p + 1− ∣∣supp( Fˆ )∣∣.
Suppose that x ∈ supp( Fˆ ). By the above, there is a subset T of S such that x + |T | ∈ supp( fˆ ) = Aˆ
and x+ |S| − |T | ∈ supp(gˆ) = Bˆ . As 0 |T | |S|,
x+ |T | ∈ Aˆ ⇒ x ∈ {p − |S|, . . . , p − 1, 0¯, . . . ,k − 1}
and
x+ |S| − |T | ∈ Bˆ ⇒ x ∈ {|B| − 2|S|, . . . , p − |S|}.
Therefore x = p − |S|, or x = r¯ for some r ∈ [|B| − 2|S|,k − 1].
If |A| + |B| p + 2|S| + 1, then k − 1 = p − |A| < |B| − 2|S|, hence supp( Fˆ ) = {p − |S|} and thus
|C | p. If |A| + |B| < p + 2|S| + 1, then
∣∣supp( Fˆ )∣∣ 1+ k − (|B| − 2|S|)= k + l − p + 2|S|
and hence
|C | p + 1− k − l + p − 2|S| = |A| + |B| − 2|S| − 1.
So (9) always holds. We are done.
Acknowledgment
The authors are grateful to the referee for his/her helpful comments.
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A variant of Tao's method with application to restricted sumsets

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