Lower bounds in some power sum problems
Lower bounds in some power sum problems
Johan Andersson∗
October 29, 2018
Abstract
We prove that for j ≥ 0 one has that
|zk|=1
ν=1,...,n2+j
(1 + j)(n − 1)
2(j + n2)
. (*)
This improves upon former estimates. Our proof will use Fejér kernels. We
also prove corresponding results for non pure power sums, and the pure
power sum estimate
− o(1)
n ≤ inf
|zk|=1
ν=1,...,⌊αn2⌋
for constants α > 1. This improves further on (*) when j ≥ 2n2.
1 Introduction
The power sum method of Turán (see Turán [Tur84] or Montgomery [Mon94]
Chapter 5) allows us to obtain lower bounds for power sums
ν=N(n),...,M(n)
|g(ν)|, (1)
where
g(ν) =
k (2)
∗Department of Mathematics, Stockholm University, SE-10691, Sweden. johana@math.su.se
http://arxiv.org/abs/0704.1879v1
for zk and bk complex numbers, and M(n) −N(n) ≥ n a function of n. We will
henceforth assume that the bk ≥ 0 are positive. In particular we are interested
in the case of pure power sums (bk = 1)
S(ν) =
zνk , (3)
and the minimum norm mink |zk| = 1. We will also assume that N(n) = 1. In
this case a number of results have been proved.
ν=1,...,n
|S(ν)| ≥ 1, (Turán [Tur69])
ν=1,...,2nm−m(m+1)+1
|S(ν)| ≥
m. (1 ≤ m ≤ n) (Andersson [And96])
Under the min norm it seems reasonable that the minimal systems (z1, . . . , zn)
which minimize the expressions actually lies on, or very close to the unit circle.
This has been difficult to prove and in fact in that case, when the zk are uni-
modular, Newman, Cassels and Szalay have independently proved the stronger
result
1≤ν≤cn
cn− n+ 1
. ([Tur84], Theorem 7.3)
2 One sided bounds
We will denote
g(ν) =
bke(θkν), (4)
where θk are real numbers and bk > 0. We will let
A = g(0) =
bk, and B =
In this section we will furthermore assume that g is real valued. In particular
this implies that
g(ν) = g(−ν). (5)
We will also let
g+(ν) =
g(ν), g(ν) > 0,
0, otherwise,
and g−(ν) =
g(ν), g(ν) < 0,
0, otherwise.
It is clear that
g(ν) = g+(ν) + g−(ν),
|g(ν)| = g+(ν)− g−(ν). (6)
Our method of proof will use the Fejér kernel
Fm+1(x) =
1− |ν|
e(νx). (7)
The Fejér kernel can be written as
Fm+1(x) =
sin π(m+ 1)x
sin πx
, (8)
and is thus non negative. We will let
g+(ν)>0
1≤ν≤m
, and β =
g−(ν)<0
1≤ν≤m
. (9)
From the representation (8) it follows that Fm+1(0) = m+1. From this it is clear
, (10)
and thus also
α + β ≤ 1. (11)
2.1 The first method
Lemma 1. One has that
|g(ν)|2 ≥ (m+ 1)B − A
Proof. We have that
1− |ν|
|g(ν)|2 =
k,l=1
bkblFm+1(θk − θl),
which by the contribution of the diagonal k = l, and the non negativity of the
Fejér kernel, eq. (8) implies that
|g(ν)|2 ≥
b2kFm+1(0).
The result follows by subtracting the term ν = 0 and using equation (5).
Lemma 2. Suppose that |g(v)| ≤ M for ν = 1, . . . , m. Then
g+(ν) ≥
B(m+ 1)− AM − A2
Proof. Since g+(ν) = (|g(ν)| + g(ν))/2 and obviously |g(ν)| ≥ |g(ν)|2/M when
ν 6= 0 we have that
g+(ν) ≥
|g(ν)|2
+ g(ν)
The first term can be estimated by Lemma 1 and gives the contribution
B(m+ 1)− A2
The second term can be investigated by use of the Fejér kernel. By the non
negativity of the Fejér kernel, equation (8) we have that
1− |ν|
g(ν) ≥ 0
By the fact that g(0) = A and using equation (5) we find that
g(ν) ≥ −A
, (12)
which gives the remaining contribution to our Lemma.
We now prove the following Theorem.
Theorem 1. Suppose that |g(ν)| ≤ M for ν = 1, . . . , m. Then one has that
ν=1,...,m
g+(ν) ≥
B(m+ 1)−AM − A2
Proof. This follows from Lemma 2 and equation (10).
2.2 An improvement for large values of m
As m tends to infinity Theorem 1 will give us
ν=1,...,m
g+(ν) ≥ B
− o(1),
We will prove a stronger results which allows us to obtain
ν=1,...,m
g+(ν) ≥ M + 2−
− o(1).
This will give sharper results when M ≍
B and for large m.
Theorem 2. Suppose that |g(ν)| ≤ M for ν = 1, . . . , m. If B(m + 1) − A2 −
mM2 ≥ 0 then one has that
ν=1,...,m
g+(ν) ≥ B(m+ 1)− A
In case B(m+ 1)−A2 −mM2 ≤ 0 one has that
ν=1,...,m
g+(ν) ≥ M + 2×
B(m+ 1)− A2 −mM2
B(m+ 1)− A2 − AM
under the assumption that the denominator in the last fraction is positive.
Proof. By equations (9) and (11) it is clear that
g−(ν) ≥ −M(1 − α). (13)
By equation (6) and the fact that |g(ν)| ≤ M we get the inequality
g+(ν) ≥ |g(ν)|
+ g−(ν).
By combining this with equation (13) we see that
g+(ν) ≥ 2
|g(ν)|2
−M(1 − α).
which by Lemma 1 can be estimated by
B(m+ 1)−A2
−M(1 − α).
This together with the definition of α, equation (9) implies that
g+(ν) ≥
B(m+ 1)−A2
−M(1− α)
B(m+ 1)−A2 −mM2
for some ν = 1, . . . , m. We see that if B(m+1)−A2−mM2 ≥ 0 then the function
is decreasing in α and the minimum over 0 < α ≤ 1 is attained for α = 1. This
gives us case 1. In the case when B(m + 1) − A2 − mM2 ≤ 0 the function is
increasing in α and we use the following estimate
B(m+ 1)− AM −A2
which follows from Lemma 2 to obtain a lower bound. By putting this value in
the right hand side of (14) we obtain a lower bound and we obtain the second
part of our theorem. We remark that we also need that B(m+ 1)−AM −A2 is
positive since otherwise we get an α < 0.
3 A lower bound for power sums
We will now use our one sided theorems to obtain improved lower bounds for the
absolute values of power sums.
Theorem 3. Let
bνk, A = B
1 − B2, and B = B22 − B4.
Then one has that
ν=1,...,m
|g(ν)|2 ≥ B2 +
B(1 + 1/m)
AB2 + A
. (15)
One also has that
ν=1,...,m
|g(ν)|2 ≥ 2B2 − 2×
A2 − B +mB4
B(m+ 1)− AB2 − A2
when m ≥ (B −A2)/B4 and both the numerator and the denominator in the last
fraction is positive (this is true for m sufficiently large).
Proof. Let g(ν) be defined by equation (4). Then
|g(ν)|2 =
b2k +
cke(λkν),
= B2 + h(ν)
where ck = bibj and λk = θi − θj for i 6= j. It is clear that h(ν) is real valued and
hence we can use the methods of section 2.
Let us now assume that |h(ν)| ≤ B2 for ν = 1, . . . , m. By using Theorem 1
with M = B2 we have that there exist a ν with ν = 1, . . . , m such that
g+(ν) ≥ B(m+ 1)− AB2 −A
This implies (15) in case |h(ν)| ≤ B2.
We have by the definition of B that
B(m+ 1)− A2 −mB22 = B − A2 −mB4
which is negative if m ≥ (B − A2)/B4, and it follows from Theorem 2 with
M = B2 that
h(ν) ≥ B2 −
2A2 − 2B + 2mB4
B(m+ 1)−AB2 − A2
for some ν = 1, . . . , m. This implies (16) in case |h(ν)| ≤ B2.
Let us now assume that |h(ν)| > B2 for some ν = 1, . . . , m. Since |g(ν)|2 =
B2 + h(ν) ≥ 0 this means that h(ν) > B2 and |g(ν)|2 ≥ 2B2. We see that this
implies (15) since 2B2 ≥ B2+(1+1/m)B/(2B2) and the third term on the right
hand side in (15) is negative. Likewise it implies (16) since the last term on the
right hand side in (16) is negative.
4 The pure power sum case
In the pure power sum case we have that Bk = n, A = B = n
2 − n in Theorem
3 and it follows that
Corollary 1. One has that
(i) max
ν=1,...,m
|S(ν)|2 ≥ n+
(−1 + n)(1 +m− n2)
(ii) max
ν=1,...,m
|S(ν)|2 ≥ 2n− 2(1 +m− 2n
2 + n3)
(−1 + n)(1 +m− n2)
. (m > n2)
Corollary 1 (i) improves upon known results for m bigger than n2. In fact it
is convenient to write m = n2 + j and we obtain
Theorem 4. One has for j ≥ 0 that
ν=1,...,n2+j
|S(ν)| ≥
(1 + j)(n− 1)
2(j + n2)
For j = 0 and in the case of unimodular numbers zk it improves slightly on
the general lower bound
ν=1,...,n2
|S(ν)| ≥
in Turán’s problem 10 from Andersson [And96]. We obtain
Corollary 2. One has that
|zk|=1
ν=1,...,n2
|S(ν)| ≥
Another result where the lower bound follows from Theorem 4 and the upper
bound follows from Montgomery’s construction (see Montgomery [Mon94] page
101. Example 6.) is the following
Corollary 3. Suppose that n + 1 is a prime number. One then has that
2n− 1
2(n2 + n− 1)
≤ inf
|zk|=1
ν=1,...,n2+n−1
|S(ν)| ≤
n+ 1.
We remark that the lower bound holds in general. We see that this approxi-
mately halves the previous gap between the upper and lower bound. For further
discussions of explicit constructions that yields similar upper bounds in power
sum problems, see our paper Andersson [Anda].
In our paper Andersson [Andb] page 17, we considered functions Λ that fulfills
n(Λ(α)− o(1)) ≤ inf
|zk|=1
ν=1,...,⌊αn2⌋
|S(ν)|.
We proved that we can choose Λ(α) = 1 for α > 0 and furthermore that for
0 < α ≤ 1 we proved that it is the best possible. We asked whether the function
must be identically 1 or must be bounded. While we can not answer if there
exist such an unbounded function it follows from Corollary 1 that we can choose
a Λ such that limα→∞ Λ(α) =
2. More specifically we obtain the following
Theorem.
Theorem 5. Let α ≥ 1 be a constant. One then has that
Φ(α)− o(1)
n ≤ inf
|zk|=1
ν=1,...,⌊αn2⌋
|S(ν)| ≤
⌈α⌉+ o(1)
where
Φ(α) =
, 1 ≤ α ≤ 3,
, α ≥ 3.
Proof. The lower bound for 1 ≤ α ≤ 3 follows from Corollary 1 (i) with m =
⌊αn2⌋. The lower bound for 3 ≤ α follows from Corollary 1 (ii) with m = ⌊αn2⌋.
The upper bound follows from Theorem 6 in Andersson [Andb].
In particular this will give us
− o(1)
n ≤ inf
|zk|=1
ν=1,...,2n2
|S(ν)| ≤
2 + o(1)
We see that the lower and upper bounds are not the same and we do not yet have
the true asymptotic. This contrasts to the case when we take the maximum over
the interval ν = 1, . . . , n2 where we proved (see Andersson [Andb])
|zk|=1
ν=1,...,n2
|S(ν)| ∼
References
[Anda] Johan Andersson. Explicit solutions to certain inf max problems from
Turán power sum theory, arXiv:math/0607238v2 [math.NT]. To appear
in Indagationes Mathematicae.
[Andb] Johan Andersson. Turán’s problem 10 revisited, arXiv:math/0609271v1
[math.NT].
[And96] Johan Andersson. On some power sum problems of Turán and Erdős.
Acta Math. Hungar., 70(4):305–316, 1996.
[Mon94] Hugh L. Montgomery. Ten lectures on the interface between analytic
number theory and harmonic analysis, volume 84 of CBMS Regional
Conference Series in Mathematics. Published for the Conference Board
of the Mathematical Sciences, Washington, DC, 1994.
[Tur69] Paul Turán. On a certain limitation of eigenvalues of matrices. Aequa-
tiones Math., 2:184–189, 1969.
[Tur84] Paul Turán. On a new method of analysis and its applications. Pure and
Applied Mathematics (New York). John Wiley & Sons Inc., New York,
1984. With the assistance of G. Halász and J. Pintz, With a foreword
by Vera T. Sós, A Wiley-Interscience Publication.
	Introduction
	One sided bounds
	The first method
	An improvement for large values of m
	A lower bound for power sums
	The pure power sum case
