Table Of ContentCANCELLATIONS AMONGST KLOOSTERMAN SUMS
6 IGOR E. SHPARLINSKI AND TIANPING ZHANG
1
0
2 Abstract. We obtain several estimates for bilinear form with
Kloosterman sums. Such results can be interpreted as a measure
n
of cancellations amongst with parameters from short intervals. In
a
J particular,forcertainrangesofparametersweimprovesomerecent
2 results of Blomer, Fouvry, Kowalski, Michel, and Mili´cevi´c (2014)
2 and Fouvry, Kowalskiand Michel (2014).
]
T
N
.
h
t 1. Introduction
a
m
Let p be a sufficiently large prime. For integers m and n we define
[
the Kloosterman sum
2
v p−1
3 K (m,n) = e (mx+nx ),
p p p
2
1 Xx=1
5
where x is the multiplicative inverse of x modulo p and
0 p
.
1
e (z) = exp(2πiz/p).
0 p
6
1 Furthermore, given two intervals
:
v
i I = [K +1,K +M], J = [L+1,L+N] ⊆ [1,p−1],
X
r
a and two sequences of weights A = {α } and B = {β } , we
m m∈I n n∈J
define the bilinear sums of Kloosterman sums
S (A,B;I,J) = α β K (mn,1).
p m n p
m∈In∈J
XX
2010 Mathematics Subject Classification. 11D79, 11L07.
Key words and phrases. Kloosterman sums, cancellation, bilinear form.
This work was supported in part by ARC Grant DP140100118 (for I. E. Sh-
parlinski),by NSFC Grant11201275and the Fundamental ResearchFunds for the
Central Universities Grant GK201503014(for T. P. Zhang).
T. P. Zhang was the corresponding author ([email protected]).
1
2 I. E. SHPARLINSKIAND T. P. ZHANG
We also consider the following special cases
S (A;I,J) = S A,{1}N ;I,J = α K (mn,1),
p p n=1 m p
m∈In∈J
(cid:0) (cid:1) XX
S (I,J) = S {1}M ,{1}N ;I,J = K (mn,1),
p p m=1 n=1 p
m∈In∈J
(cid:0) (cid:1) XX
S (I) = S {1}M ,∅;I,J = K (m,1).
p p m=1 p
m∈I
(cid:0) (cid:1) X
Making the change of variable x → nx (mod p), one immediately
observes that K (mn,1) = K (m,n), thus we also have
p p
S (I,J) = K (m,n).
p p
m∈In∈J
XX
We aslo define, for real σ > 0,
1/σ 1/σ
kAk = |α |σ and kBk = |β |σ
σ m σ n
! !
m∈I n∈J
X X
with the usual convention
kAk = max|α | and kBk = max|β |.
∞ m ∞ n
m∈I n∈J
By the Weil bound we have
|K (m,n)| ≤ 2p1/2,
p
see [7, Theorem 11.11]. Hence
(1.1) S (A,B;I,J) ≤ 2kAk kBk p1/2.
p 1 1
Weare interested in studying cancellations amongst Kloostermansums
and thus improvements of the trivial bound (1.1).
Throughout the paper, as usual A ≪ B is equivalent to the inequal-
ity |A| ≤ cB with some constant c > 0 (all implied constants are
absolute throughout the paper).
2. Previous results
First we note that by a very special case of a much more general
result of Fouvry, Michel, Rivat and S´ark¨ozy [4, Lemma 2.3] we have
S (I) ≪ plogp
p
which for M ≥ p1/2logp improves the trivial bound S (I) ≤ 2Mp1/2
p
following from (1.1). Recently, Fouvry, Kowalski, Michel, Raju, Rivat
and Soundararajan [5, Corollary 1.6] have given the bound
S (I) ≪ Mp1/2(logp)−η
p
CANCELLATIONS AMONGST KLOOSTERMAN SUMS 3
provided that M ≥ p1/2(logp)−η with some absolute constant η > 0.
It is also easy to derive from [13, Theorem 7] that
S (I,J) ≪ MNp1/4 +M1/2N1/2p1+o(1),
p
which improves the trivial bound from (1.1) for MN ≥ p1+ε for any
fixed ε > 0.
The sums S (A,B;I,J) and S (A;I,J) have been estimated by
p p
Fouvry, Kowalski and Michel [3, Theorem 1.17] as a part of a much
more general result about sums of so-called trace functions. Then,
by[3, Theorem1.17(2)],forinitialintervals I = [1,M] and J = [1,N],
we have
(2.1) S (A;I,J) ≤ kAk p1+o(1).
p 1
Furthermore, by a result of Blomer, Fouvry, Kowalski, Michel, and
Mili´cevi´c[1, Theorem6.1], alsoforaninitialinterval I andanarbitrary
interval J , with
MN ≤ p3/2 and M ≤ N2
we have
(2.2) S (A;I,J) ≤ (kAk kAk )1/2M1/12N7/12p3/4+o(1).
p 1 2
One can also find in [1, 3, 10] a series of other bounds on the sums
S (A;I,J) and S (A,B;I,J) and also on more general sums.
p p
Finally, Khan [9] has given a nontrivial estimate for the analogue
of S (I) modulo a fixed prime power which is nontrivial already for
p
M ≥ pε.
3. New results
We start with the sums S (I,J) and present a bound which im-
p
proves (1.1) already for MN ≥ p1/2+ε.
Theorem 3.1. We have,
S (I,J) ≪ (p+MN)po(1).
p
We now estimate S (A;I,J).
p
Theorem 3.2. We have,
S (A;I,J) ≪ kAk N1/2p.
p 2
We can re-write the bounds (2.1) and (2.2) in terms of the kAk as
∞
(3.1) S (A;I,J) ≪ kAk Mp1+o(1)
p ∞
and
(3.2) S (A;I,J) ≪ kAk M5/6N7/12p3/4+o(1),
p ∞
4 I. E. SHPARLINSKIAND T. P. ZHANG
respectively, and the bound of Theorem 3.2 as
(3.3) S (A;I,J) ≪ kAk M1/2N1/2p1+o(1).
p ∞
Wenowseeforanyfixed ε > 0 thebound(3.3)improves(3.1)and(3.2)
for
N < Mp−ε and M4N ≥ p3+ε
respectively, and also applies to intervals I and J at arbitrary posi-
tions.
4. Preparations
We need the following simple result.
Lemma 4.1. For any integers X and Y with 1 ≤ X,Y < p, the con-
gruence
xy ≡ 1 (mod p), 1 ≤ |x| ≤ X, 1 ≤ |y| ≤ Y
has at most (XY/p+1)po(1) solutions.
Proof. Writing xy ≡ 1 (mod p) as xy = 1 + kp for some integer k
with |k| ≤ XY/p and using the bound on the divisor function, see [7,
Equation (1.81)], we get the desired estimate. ⊔⊓
We also need the following well-known result, which dates back to
Vinogradov [15, Chapter 6, Problem 14.a].
Lemma 4.2. For arbitrary set U,V ⊆ {0,...,p−1} and complex num-
bers ϕ and ψ with
u v
|ϕ |2 ≤ Φ and |ψ |2 ≤ Ψ,
u v
u∈U v∈V
X X
we have
ϕ ψ e (uv) ≤ ΦΨp.
u v p
(cid:12) (cid:12)
(cid:12)Xu∈U Xv∈V (cid:12) p
(cid:12) (cid:12)
(cid:12) (cid:12)
5. Proof of Theorem 3.1
(cid:12) (cid:12)
For an integer u we define
kuk = min|u−kp|
p
k∈Z
as the distance to the closest integer. Then, changing the order of
summation, we obtain
p−1
p p
S (I,J) ≪ min M, min N, ,
p
kxk kx k
x=1 (cid:26) p(cid:27) (cid:26) p p(cid:27)
X
CANCELLATIONS AMONGST KLOOSTERMAN SUMS 5
see [7, Bound (8.6)]. We now write
(5.1) S (I,J) ≪ MNS +MpS +NpS +p2S ,
p 1 2 3 4
where
p−1 p−1
1
S = 1, S = ,
1 2
kx k
p p
x=1 x=1
kxkXp≤p/M kxkXp≤p/M
kxpkp≤p/N kxpkp>p/N
p−1 p−1
1 1
S = , S = .
3 4
kxk kxk kx k
p p p p
x=1 x=1
kxkXp>p/M kxkXp>p/M
kxpkp≤p/N kxpkp>p/N
By Lemma 4.1 we immediately obtain
(5.2) S ≤ (p/MN +1)po(1).
1
To estimate S , we define I = ⌈logp⌉ and write
2
I
S ≤ S ,
2 2,i
i=0
X
where
p−1
1
S =
2,i
kx k
p p
x=1
kxkXp≤p/M
ei+1p/N≥kxpkp>eip/N
p−1 p−1
≪ e−iNp−1 1 ≪ e−iNp−1 1.
x=1 x=1
kxkXp≤p/M kxkXp≤p/M
ei+1p/N≥kxpkp>eip/N kxpkp≤ei+1p/N
Now we use Lemma 4.1 again to derive
I
S ≤ M−1 +e−iNp−1 po(1)
2
i=0
X(cid:0) (cid:1)
I
(5.3) ≪ (I +1)M−1po(1) +Np−1+o(1) e−i
i=0
X
≪ (I +1)M−1po(1) +Np−1+o(1)
≪ M−1po(1) +Np−1+o(1).
Similarly we obtain
(5.4) S ≤ N−1po(1) +Mp−1+o(1).
3
6 I. E. SHPARLINSKIAND T. P. ZHANG
Finally, we write
I
S ≤ S ,
4 4,i,j
i,j=0
X
where
p−1
1
S =
4,i,j
kxk kx k
p p p
x=1
ei+1p/M≥Xkxkp>eip/M
ej+1p/N≥kxpkp>ejp/N
p−1
≪ e−i−jMNp−2 1
x=1
ei+1p/M≥Xkxkp>eip/M
ej+1p/N≥kxpkp>ejp/N
p−1
≪ e−i−jMNp−2 1.
x=1
kxkp≤Xei+1p/M
kxpkp≤ej+1p/N
Applying Lemma 4.1 one more time, we obtain
S ≪ e−i−jMNp−2 ei+jp/MN +1 po(1)
4,i,j
= (p−1 +e−i−jM(cid:0)Np−2)po(1). (cid:1)
Hence
I
S ≤ (p−1 +e−i−jMNp−2)po(1)
4
i,j=0
(5.5) X
≤ (I +1)2p−1+o(1) +MNp−2+o(1)
≤ p−1+o(1) +MNp−2+o(1).
Combining (5.2), (5.3), (5.4) and (5.5) we obtain the result.
6. Proof of Theorem 3.2
Changing the order of summation, as in the proof of Theorem 3.1
we obtain
p−1
S (A;I,J) = α γ e (mx ) ,
p m x p p
(cid:12) (cid:12)
(cid:12)mX∈IXx=1 (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) p (cid:12)
|γ | ≤ min N, .
x
kxk
(cid:26) p(cid:27)
CANCELLATIONS AMONGST KLOOSTERMAN SUMS 7
Thus, similarly to the proof of Theorem 3.1 we define I = ⌈logp⌉
and write
I
(6.1) S (I,J) ≪ |S |+ |S |,
p 0 i
i=1
X
where
p−1
S = α γ e (mx ),
0 m x p p
m∈I x=1
XkxkXp≤p/N
p−1
S = α γ e (mx ), i = 1,...,I.
i m x p p
m∈I x=1
Xei+1p/N≥Xkxkp>eip/N
Now use Lemma 4.2, we have
(6.2) |S | ≪ kAk N (p/N +1)p ≤ kAk N1/2p.
0 2 2
Also, for i = 1,...,I, usingpthat if ei+1p/N ≥ kxkp > eip/N then
γ ≪ Ne−i, hence, by Lemma 4.2, we obtain
x
p−1
S = α γ e (mx )
i m x p p
m∈I x=1
Xei+1p/N≥Xkxkp>eip/N
≪ kAk (N2e−2ieip/N)1/2p1/2 = e−i/2kAk N1/2p.
2 2
Therefore,
I I
(6.3) |S | ≪ kAk N1/2p e−i/2 ≪ kAk N1/2p.
i 2 2
i=1 i=1
X X
Combining (6.2) and (6.3), we obtain the result.
7. Comments
It is also natural to consider cancellations between some other ex-
ponential and character sums. For example, in [14] one can find some
bound on the following sums
S (f,A,B;C) = α β e (v/f(u)),
p u v p
(u,v)∈C
X
T (f,A,B;C) = α β χ(v +f(u)),
p u v
(u,v)∈C
X
(where χ is a multiplicative character modulo p), over a convex set
C ⊆ [1,U]×[1,V], with some integer 1 ≤ U,V < p.
8 I. E. SHPARLINSKIAND T. P. ZHANG
Here we also note that one can also obtain a nontrivial cancellation
for sums
H (a;I) = G (am)
k,p k,p
m∈I
X
of Gaussian sums
p−1
G (a) = e axk
k,p p
x=0
X (cid:0) (cid:1)
with a positive integer k | p−1. Indeed, we define
p−1
τ (a;χ) = χ(x)e (ax),
p p
x=1
X
where χ is a multiplicative character; we refer to [7, Chapter 3] for a
background on multiplicative characters. Then by the orthogonality of
characters, we have
G (a) = χ(a)τ (a;χ),
k,p p
χXk=χ0
χ6=χ0
where the summation is over all nonprincipal multiplicative characters
χ modulo p, such that χk is the principal character χ , see also [11,
0
Theorem 5.30]. Using that |τ (a;χ)| = p1/2 for any nonprincipal mul-
p
tiplicative characters χ and integer a with gcd(a,p) = 1, we derive
|H (a;I)| = p1/2 χ(a) χ(m).
k,p
χXk=χ0 mX∈I
χ6=χ0
Thus applying the Burgess bound, see [7, Equation (12.58)], we derive
that
H (a;I) ≪ M1−1/νp1/2+(ν+1)/(4ν2)(logp)1/ν
k,p
(7.1)
= M1−1/νp(2ν2+ν+1)/(4ν2)(logp)1/ν
for any fixed k | p−1 and ν = 1,2,....
Similarly, for general quadratic polynomials f(X) = aX2+bX, with
gcd(a,p) = 1, we can define the double sums
p−1
F (a,b;I) = e m ax2 +bx .
p p
m∈I x=0
XX (cid:0) (cid:0) (cid:1)(cid:1)
CANCELLATIONS AMONGST KLOOSTERMAN SUMS 9
It is easy to see that
p−1 b2 p−1
e (ax2 +bx) = e − e a(x+b/(2a))2
p p p
4a
x=0 (cid:18) (cid:19)x=0
X X (cid:0) (cid:1)
b2 p−1 a b2
= e − e ax2 = e − G (1),
p p p 2,p
4a p 4a
(cid:18) (cid:19)x=0 (cid:18) (cid:19) (cid:18) (cid:19)
X (cid:0) (cid:1)
where (a/p) is the Legendre symbol of a modulo p. Hence
p−1
am (bm)2
e m(ax2 +bx) = G (1) e −
p 2,p p
p 4am
m∈I x=0 m∈I(cid:18) (cid:19) (cid:18) (cid:19)
XX (cid:0) (cid:1) X
a m b2m
= G (1) e − .
2,p p
p p 4a
(cid:18) (cid:19) m∈I(cid:18) (cid:19) (cid:18) (cid:19)
X
Now, using the bound of Burgess [2] on short mixed sums (see [6, 8, 12]
for various generalisations) we easily derive that for any fixed ν =
2,3,... we have
F (a,b;I) ≪ M1−1/νp1/2+1/(4(ν−1))(logp)2
p
(7.2)
= M1−1/νp(2ν−1)/(4(ν−1))(logp)2,
where the implied constant may depend on ν.
We note that the bounds (7.1) and (7.2) are nontrivial provided that
M ≥ p1/4+ε for any fixed ε > 0 and sufficiently large p.
Acknowledgement
The second author gratefully acknowledge the support, the hospi-
tality and the excellent conditions at the School of Mathematics and
Statistics of UNSW during his visit.
This work was supported in part by ARC Grant DP140100118 (for
I. E. Shparlinski), by NSFC Grant 11201275 and the Fundamental
Research Funds for the Central Universities Grant GK201503014 (for
T. P. Zhang).
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10 I. E. SHPARLINSKIAND T. P. ZHANG
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Department of Pure Mathematics, University of New South Wales,
Sydney, NSW 2052, Australia
E-mail address: [email protected]
School of Mathematics and Information Science, Shaanxi Normal
University, Xi’an 710062 Shaanxi, P. R. China
E-mail address: [email protected]
