NOTE ON A LOWER BOUND OF THE WEYL SUM IN BOURGAIN’S
NLS PAPER (GAFA ’93)
TADAHIRO OH
1. Introduction
In this note, we go over Bourgain’s counterexample [1] to the periodic L6 -Strichartz
estimate for the Schrödinger equation on T. In [1], Bourgain proved the periodic L6 Strichartz estimate with a slight loss of derivative:
X
2πi(nx+n2 t) ≤ CN {an }ℓ2
,
(1.1)
an e
|n|≤N
L6 (T2 )
|n|≤N
where the constant CN is bounded above by
log N
c log
log N
≪ N ε,
CN . e
for any ε > 0.
(1.2)
The proof is based on a simple divisor counting argument, and the loss (1.2) basically comes
from the number of divisors of an integer N .
In the same paper, he also showed that some loss of derivative in (1.1) was indeed
necessary. More precisely, it was shown that
1
CN & (log N ) 6
(1.3)
for the initial condition an = χ[0,N ] (n).1 The main part of the argument is based on
establishing the following (lower) bound on the Weyl sum:
X
N 2πi(nx+n2 t) N
∼ √
e
.
q
(1.4)
n=0
for fixed x and t in the major arc M0 (q, a, b).2 See Proposition 3.1. (Also, see Theorem
2.3.) Here, the major arc M0 (q, a, b) is defined for q, a, and b satisfying
1
1 ≤ a < q < N 2,
(a, q) = 1,
0 ≤ b < q,
(1.5)
and is given by
M0 (q, a, b) =
(x, t) ∈ [0, 1]2
b 1
: x − ≤
, t −
q
100N
1
a ≤
.
q
100N 2
(1.6)
1In fact, Bourgain’s argument in [1] yields (1.3) modulo a log log N -factor. See (4.3) below.
2In the application of the Hardy-Littlewood circle method, one often divides the sum into dyadic blocks
and define major and minor arcs for each dyadic block. Here, we do not need such a dyadic decomposition.
1
2
TADAHIRO OH
2. Dirichlet’s theorem, Gauss sum, and Weyl sum
Recall the following theorem:
Theorem 2.1 (Dirichlet). Let θ ∈ [0, 1] and N ≥ 1. Then, there exist integers a and q
satisfying 1 ≤ a ≤ q ≤ N and (a, q) = 1 such that
θ − a ≤ 1 ,
(2.1)
q qN
where k · k denotes the distance to the closest integer.
Proof. Consider the N + 1 numbers jθ (mod 1) for j = 0, 1, . . . , N . By the pigeon hole
principle, there exist two distinct integers m, n ∈ {0, 1, · · · , N } with m > n such that
|mθ − nθ − a′ | ≤
1
N
(2.2)
for some non-negative integer a′ . Let q ′ = m − n. If 1 ≤ a′ ≤ q ′ and (a′ , q ′ ) = 1, then (2.1)
holds with a = a′ and q = q ′ after dividing (2.2) by q ′ . It remains to consider the following
cases.
(a) If a′ = 0, then it follows from (2.2) that |θ| ≤ N1 . Hence, (2.1) holds with a = q = 1.
(b) If a′ > q, then from (2.2), we have N1 ≥ a′ − q ′ θ ≥ q ′ (1 − θ). Once again, (2.1) holds
with a = q = 1.
(c) If (a′ , q ′ ) 6= 1 (but 1 ≤ a′ ≤ q ′ ≤ N ), then we can write a′ = ka and q ′ = kq for some
′
1
≥ q′1N ≥ |θ − aq′ | ≥ |θ − aq |.
k ≥ 2 such that (a, q) = 1. Then, from (2.2) we obtain qN
Hence, (2.1) holds in this case as well.
Next, we recall the estimate on the Gauss sum. Given positive integers a and q with
(a, q) = 1, the Gauss sum S(a, q) is defined by
S(a, q) :=
q
X
2a
q
e2πin
(2.3)
.
n=1
More generally, for a, q ∈ N and b ∈ Z with (a, q) = 1, we can define the Gauss sum
S(a, b, q) by
S(a, b, q) :=
q
X
2πi( aq n2 + qb n)
e
.
(2.4)
n=1
i.e. S(a, q) = S(a, 0, q).
Theorem 2.2 (Gauss sum). Let a, q ∈ N and
following holds for the Gauss sums:
(a) When b is even,
√

if
 q,
|S(a, b, q)| = 0,
if

√
2q, if
b ∈ Z such that (a, q) = 1. Then, the
q is odd,
q ≡ 2 (mod 4),
q ≡ 0 (mod 4),
(2.5)
(b) When b is odd,
√

√q,
|S(a, q)|, |S(a, b, q)| =
2q,


0,
if q is odd,
if q ≡ 2 (mod 4),
if q ≡ 0 (mod 4),
(2.6)
A LOWER BOUND ON THE WEYL SUM
3
Proof. First, note that the Gauss sum (2.4) is invariant if we shift the range of summation.
Thus, we have
2
|S(a, b, q)| = S(a, b, q)S(a, b, q) =
=
q X
q
X
q
q X
X
2πi( aq (m2 −n2 )+ qb (m−n))
e
n=1 m=1
2πi( aq ((n+ℓ)2 −n2 )+ qb ((n+ℓ)−n))
e
n=1 ℓ=1
=
q
q X
X
ℓ=1
2πi(2ℓ aq )n
e
n=1
2πi( aq ℓ2 + qb ℓ)
.
e
Here, the inner sum is 0 unless
2ℓa ≡ 0 (mod q).
(2.7)
If (2.7) holds, the inner sum is equal to q.
• Case 1: Suppose that q is odd. Since (a, q) = 1 = (2, q), it follows from (2.7) that ℓ = q.
Thus, we have
|S(a, b, q)|2 = q.
• Case 2: Next, suppose that q ≡ 2 (mod 4). Since (a, q) = 1, we have 2ℓ ≡ 0 (mod q).
Namely, ℓ = 2q or q. Thus, we have
(
0,
if b is even,
+b)
2πi(qa+b)
2
πi( qa
+e
)=
|S(a, b, q)| = q(e 2
2q, if b is odd.
• Case 3: Lastly, suppose that q ≡ 0 (mod 4). In this case,
we have
(
qa
2q,
|S(a, b, q)|2 = q(eπi( 2 +b) + e2πi(qa+b) ) =
0,
qa
2
is an even number. Thus,
if b is even,
if b is odd.
This proves (2.5) and (2.6).
Lastly, we state the classical estimate on the Weyl sum.
Theorem 2.3 (Weyl sum). Let x, t ∈ R and a and q be integers with (a, q) = 1 such that
t − a ≤ 1 .
q q2
Then, the following bound holds:
X
N 2πi(nx+n2 t) 1
1
N
.
2
(log q) 2 .
e
1 + q
q2
n=0
Remark 2.4. In general, given a polynomial p(n) of degree k, we have
X
1
N 2πip(n)t 1
1
q 2k−1
1+ε
e
+ + k
.
≤ Cε,k N
N
q N
n=0
(2.8)
4
TADAHIRO OH
3. Lower bound (1.4)
In this section, we prove the lower bound (1.4) under certain conditions on q and b. The
basic idea is to use the bound on the Gauss sum (Theorem 2.2) after replacing a certain
summation by integration (See (3.3).)
Proposition 3.1. Let q, a, and b be as in (1.5). Then, for (x, t) ∈ M0 (q, a, b), we have
the following lower bound on the Weyl sum:
X
N 2πi(nx+n2 t) N
& √
,
(3.1)
e
q
n=0
provided that one of the following conditions holds:
(a) q is odd,
(b) q ≡ 0 (mod 4) and b is even, or
(c) q ≡ 2 (mod 4) and b is odd.
Remark 3.2. The following proof does not tell us what happens (i) when q ≡ 2 (mod 4)
and b is even, and (ii) q ≡ 0 (mod 4) and b is odd.
Proof. Let α = t −
N
X
a
q
and β = x − qb . By writing n = mq + ℓ with 1 ≤ ℓ ≤ q, we have
[N
]q
q
2πi(nx+n2 t)
e
=
X
2 t)
e2πi(nx+n
+ O(q)
n=1
n=0
=
[N ]
q X
q
X
2πi{(mq+ℓ)( qb +β)+(mq+ℓ)2 ( aq +α)}
e
+ O(q)
ℓ=1 m=0
=
q
X
[N ]
2πi(aℓ2 +bℓ)/q
e
q
X
e2πi{(mq+ℓ)
2 α+(mq+ℓ)β}
+ O(q),
(3.2)
m=0
ℓ=1
since mq + ℓ ≡ ℓ (mod q) and (mq + ℓ)2 ≡ ℓ2 (mod q). Note that the error O(q) in (3.2)
1
1
3
is acceptable since O(q) . N 2 ≪ N 4 < √Nq under the assumption q < N 2 .
The first sum on the right hand side of (3.2) is basically the Gauss sum. However, we
can not use Theorem 2.2 since the inner sum also depends on ℓ. Thus, we first need to
replace the inner sum by an integral and get rid of the
(i.e. van der Corput
ℓ-dependence.
N
approximation lemma-type argument.) Fix m ∈ Z ∩ 0, [ q ] . Then, for y ∈ [m, m + 1], we
have
{(mq+ℓ)2 α + (mq + ℓ)β} − {(yq + ℓ)2 α + (yq + ℓ)β}
= ((m + y)q + 2ℓ)(m − y)qα + (m − y)qβ ≤
1
1
20N 2
.
Hence, by Mean Value Theorem, we have
[N ]
q
X
2πi{(mq+ℓ)2 α+(mq+ℓ)β}
e
=
m=0
=
ˆ
ˆ
[N
]+1
q
2πi{(yq+ℓ)2 α+(yq+ℓ)β}
e
dy + O
0
N
q
0
2πi{(yq+ℓ)2 α+(yq+ℓ)β}
e
dy + O
N 21 q
N 21 q
.
(3.3)
A LOWER BOUND ON THE WEYL SUM
1
5
1
This error O( Nq2 ) becomes O(N 2 ) under the ℓ-summation in (3.2). Note that this is an
acceptable error as before. By change of variables z = yq + ℓ (for fixed ℓ), the integral on
the right hand side of (3.3) becomes
ˆ
ˆ
ˆ N
ℓ
q
1 N +ℓ 2πi(z 2 α+zβ)
1 N 2πi(z 2 α+zβ)
2πi{(yq+ℓ)2 α+(yq+ℓ)β}
e
dy =
e
dz =
e
dz + O
q ℓ
q 0
q
0
| {z }
O(1)
N
N
1
2
(e2πi(z α+zβ) − 1)dz + O(1)
+
q
q 0
2πN N
+O
+ O(1),
=
q
50q
=
ˆ
(3.4)
where we used Mean Value Theorem in the last equality. The error O(1) in (3.4) becomes
O(q) under the ℓ-summation in (3.2), which is again acceptable.
Finally, the estimate (3.1) follows from Theorem 2.2 with (3.2), (3.3), and (3.4), provided
one of the following conditions holds: (a) q is odd, (b) q ≡ 0 (mod 4) and b is even, or (c)
q ≡ 2 (mod 4) and b is odd.
4. Proof of (1.3)
In this section, we complete the construction of the counterexample to the periodic L6 Strichartz estimate. Namely, we prove (1.3) modulo an log log N -factor. Define fN by
fN (x, t) =
N
X
2 t)
e2πi(nx+n
.
n=0
1
Then, kfN (·, 0)kL2x (T) = N 2 .
Fix q, a and b satisfying (1.5). Then, from Proposition 3.1, we have
ˆ
N3
|fN (x, t)|6 dxdt & 3 ,
q
M0 (q,a,b)
(4.1)
(at least) when q is odd.
Lemma 4.1 (Disjointness of the major arcs). The major arcs defined in (1.6) are disjoint.
More precisely, let q, a, b and q ′ , a′ , b′ satisfy (1.5), respectively. Suppose that M0 (q, a, b) ∩
M0 (q ′ , a′ , b′ ) 6= ∅. Then, M0 (q, a, b) = M0 (q ′ , a′ , b′ ), i.e. q = q ′ , a = a′ and b = b′ .
Proof. Suppose that (x, t) belongs to two major arcs, i.e. (x, t) ∈ M0 (q, a, b)∩M0 (q ′ , a′ , b′ ),
where q, a, b and q ′ , a′ , b′ satisfy (1.5), respectively.
′
If aq 6= aq′ , then we have
1
a′ |aq ′ − a′ q|
a 1
1
> t − + t − ′ ≥
≥ ′ > .
2
′
50N
q
q
qq
qq
N
This is clearly a contradiction. Now, suppose that q = q ′ and a = a′ , but b 6= b′ .
b′ |b − b′ |
b 1
1
1
≥ x − + x − ′ ≥
≥ > 1.
50N
q
q
q
q
N2
This is again a contradiction.
6
TADAHIRO OH
By Lemma 4.1 and (4.1), we have
1
ˆ
T2
|fN (x, t)| dxdt ≥
2 −1
NX
≥
2 −1
NX
6
q=1
q
q ˆ
X
X
|fN (x, t)|6 dxdt
q ˆ
q
X
X
|fN (x, t)|6 dxdt
a=1 b=1
(a,q)=1
1
a=1 b=1
q=1
q odd (a,q)=1
M0 (q,a,b)
M0 (q,a,b)
1
&
2 −1
NX
ϕ(q)
q=1
q odd
N3
,
q2
where ϕ(q) is Euler’s function representing the number of positive integers not greater than
and relatively prime to q.
Theorem 328 in Hardy-Wright [2] states the following:
ϕ(n) log log n
= e−γ ,
n
Pn 1
where γ is Euler’s constant given by γ = limn→∞
k=1 k − log n = 0.5772 · · · . Moreover,
it is known that
n
(4.2)
ϕ(n) ≥ γ
3
e log log n + log log
n
for n ≥ 3. Hence, we have
lim inf
n→∞
1
ˆ
T2
|fN (x, t)|6 dxdt & N 3
2 −1
NX
q=3
q odd
1
log N
& N3
.
q(log log q + 3)
log log N
(4.3)
In the last step, we crudely estimated the sum by replacing log log q by log log N . In any
case, it seems that some loss of N is inevitable,3 i.e. (1.3) does not hold as it is written in
[1].
References
[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to
nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156.
[2] G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, Fifth edition. The Clarendon
Press, Oxford University Press, New York, 1979. xvi+426 pp.
[3] H. Takaoka, N. Tzvetkov On 2D nonlinear Schrödinger equations with data on R × T, J Funct. Anal.
182 (2001), 427–442.
3Note that this is still more accurate than the computation presented in Takaoka-Tzvetkov [3], where
the summation is only over q prime.