Báo cáo toán học: "Some inequalities in functional analysis, combinatorics, and probability theory" pdf

Some inequalities in functional analysis,
combinatorics, and probability theory
Chunro ng Feng
∗
Liangpan Li
†
Jian Shen
‡
Submitted: Aug 21, 2009; Accepted: Mar 30, 2010; Published: Apr 5, 2010
Mathematics Subject Classifi cation: 46C05, 05A20, 60C05, 11T99
Abstract
The main purpose of this paper is to show that many inequalities in functional
analysis, probability theory and combinatorics are immediate corollaries of the best
approximation theorem in inner product spaces. Besides, as applications of the
de Caen-Selberg inequality, the finite field Kakeya and Nikodym problems are also
studied.
Keywords: inner product space, orthogonal projection, Kakeya set, Nikodym set
1 Brief Introduction
Let (H, < ·, · >) be an inner product space over R throughout. Given x ∈ H and a finite
dimensional subspace M, denote by x
M
the orthogonal projection of x onto M. It is
geometrically evident that (we always assume
0
0
= 0 in this paper)
x
2
 x
M


2
= max
y ∈M
< x
M
, y >
2
y
2
= max
y ∈M
< x, y >
2
y
2
. (1)
Particularly, if M = span{y
i
}
n
i=1
for some given set of elements y
1
, . . . , y
n
, then
x
2
 max
(α

1
, ,α
n
)∈R
n
< x,

n
i=1
α
i
y
i
>
2


n
i=1
α
i
y
i

2
. (2)
∗
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China & Department
of Mathematical Sciences, Loughboro ugh University, Leics, LE11 3TU, UK . E-mail:
Research was supported by the Mathematical Tianyuan Foundation of China (No. 10826090).

†
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail: lil-
Research was supported by the Mathematical Tianyuan Foundation of China
(No. 10826088).
‡
Department of Mathematics, Texas State University, San Marcos, TX 78666, USA. E-mail:
Research was supported by NSF (CNS 0835834) and Texas Higher Education Co-
ordinating Board (ARP 003615-0039-2 007).
the electronic journal of combinatorics 17 (2010), #R58 1
The main purpose of this paper is to show that many inequalities in functional analysis,
probability theory and combinatorics are immediate corollaries of (2). For the sake of
completeness we determine the unique orthogonal projection x
M
(many authors of text-
books on functional analysis only dealt the case when {y
i
}
n
i=1
are linear independent).
Write x
M
=

n
i=1
β
i
y
i

for some (β
1
, . . . , β
n
) ∈ R
n
. Since the smooth function
Ψ(α
1
, . . . , α
n
)
.
= x −
n

i=1
α
i
y
i

2
= x
2
− 2
n

i=1
α

i
< x, y
i
> +
n

i=1
n

j=1
α
i
α
j
< y
i
, y
j
>
attains its minimum d(x, M)
2
at (β
1
, . . . , β
n
),
∂Ψ
∂α
i
(β

1
, . . . , β
n
) = 0 (i = 1, 2, . . . , n).
Equivalently,





< y
1
, y
1
> < y
1
, y
2
> · · · < y
1
, y
n
>
< y
2
, y
1
> < y
2
, y

2
> · · · < y
2
, y
n
>
.
.
.
.
.
.
.
.
.
.
.
.
< y
n
, y
1
> < y
n
, y
2
> · · · < y
n
, y
n

>










β
1
β
2
.
.
.
β
n





=






< x, y
1
>
< x, y
2
>
.
.
.
< x, y
n
>





. (3)
If ( γ
1
, . . . , γ
n
) ∈ R
n
is another solution to (3), then


n


i=1
(β
i
− γ
i
)y
i


2
= (β
1
− γ
1
, · · · , β
n
− γ
n
)(< y
i
, y
j
>)
n×n



β
1
− γ

1
.
.
.
β
n
− γ
n



= (β
1
− γ
1
, · · · , β
n
− γ
n
)



0
.
.
.
0




= 0.
Consequently x
M
=

n
i=1
β
i
y
i
=

n
i=1
γ
i
y
i
.
Among many inequalities will be discussed later, we show particular interest in the de
Caen-Selberg inequality [1, 2]:



n

i=1
A

i




n

i=1
|A
i
|
2
n

j=1
|A
i
∩ A
j
|
, (4)
where {A
i
}
n
i=1
are finite sets. In Section 5 we will present some applications of the de
Caen-Selberg inequality to the study of the finite field Kakeya and Nikodym problems in
classical analysis.
the electronic journal of combinatorics 17 (2010), #R58 2

2 Inequalities in Funct i onal Analysis
2.1 Known inequalities
For any (α
1
, . . . , α
n
) ∈ R
n
, by (2) and the Cauchy-Schwarz inequality (|α
i
α
j
| 
α
2
i
+α
2
j
2
)
one obtains the Pe˘cari´c inequality [13]
x
2


n

i=1
α

i
< x, y
i
>

2
n

i=1
n

j=1
α
2
i
| < y
i
, y
j
> |
. (5)
(The following arguments are standard [13]) Substituting α
i
=
<x,y
i
>
P
n
k=1

|<y
i
,y
k
>|
into (5) yields
the Selberg inequality [1]
x
2

n

i=1
< x, y
i
>
2
n

j=1
| < y
i
, y
j
> |
. (6)
Substituting α
i
= sgn(< x, y
i

>) into (5) or applying the Cauchy-Schwarz inequality from
(6) yields the Heilbronn inequality [10]
x
2


n

i=1
| < x, y
i
> |

2
n

i=1
n

j=1
| < y
i
, y
j
> |
. (7)
The Selberg inequality (6) is certainly stronger than the Bombieri inequality [1]
x
2


n

i=1
< x, y
i
>
2
max
1in
n

j=1
| < y
i
, y
j
> |
. (8)
If {y
i
}
n
i=1
are orthogonal, then the Selberg inequality (6) turns out to be the classical
Bessel inequality
x
2

n


i=1
< x, y
i
>
2
< y
i
, y
i
>
. (9)
Substituting α
i
= 1 into (2) yields the Chung-Erd˝os inequality [3]
x
2


n

i=1
< x, y
i
>

2
n

i=1
n


j=1
< y
i
, y
j
>
. (10)
the electronic journal of combinatorics 17 (2010), #R58 3
In a partial summary,
(2) ≻ (5) ≻ (6) ≻ (7),
where (•) ≻ (••) means Estimate (•) is stronger than Estimate (••).
3 From Functional Analysis to Combinatorics
3.1 Immediate c orollaries
In this section we always choo se H = l
2
. Let A, B be finite subsets of N and χ
A
, χ
B
be
the corresponding indictor functions. Then
< χ
A
, χ
B
>= |A ∩ B|,
and χ
A
, χ

B
are orthogonal means A, B are disjoint sets. Given finite subsets {A
i
}
n
i=1
of
N, define y
i
= χ
A
i
(i ∈ [n]) and x = χ
∪
i
A
i
. Then < x, y
i
>= |(∪
j
A
j
) ∩ A
i
| = |A
i
|. By (2)
and (3 ) , we obtain
Theorem 3.1.




n

i=1
A
i



 max
(α
1
, ,α
n
)∈R
n

n

i=1
α
i
|A
i
|

2
n


i=1
n

j=1
α
i
α
j
|A
i
∩ A
j
|
=
n

i=1
n

j=1
β
i
β
j
|A
i
∩ A
j
|, (11)

where (β
1
, . . . , β
n
) ∈ R
n
is any solution to





|A
1
∩ A
1
| |A
1
∩ A
2
| · · · |A
1
∩ A
n
|
|A
2
∩ A
1
| |A

2
∩ A
2
| · · · |A
2
∩ A
n
|
.
.
.
.
.
.
.
.
.
.
.
.
|A
n
∩ A
1
| |A
n
∩ A
2
| · · · |A
n

∩ A
n
|










β
1
β
2
.
.
.
β
n





=






|A
1
|
|A
2
|
.
.
.
|A
n
|





. (12)
Note in this context the Selberg inequality (6) turns out to be the de Caen inequality
(4) and the Bessel inequality (9) turns out to be a trivial equality. Also note that
sup
α
i
>0

n


i=1
α
i
|A
i
|

2
n

i=1
n

j=1
α
i
α
j
|A
i
∩ A
j
|
= sup
α
i
>0

n


i=1
α
i
|A
i
|

2
n

i=1
n

j=1
α
2
i
|A
i
∩ A
j
|
= sup
α
i
>0
n

i=1
α

i
|A
i
|
2
n

j=1
α
j
|A
i
∩ A
j
|
.
the electronic journal of combinatorics 17 (2010), #R58 4
3.2 A slightly different variant
In this subsection, we provide a slightly different variant of (12).
Theorem 3.2. The following matrix equation always has a solution

|A
i
∩ A
j
|
|A
i
||A
j

|

n×n





q
1
q
2
.
.
.
q
n





=





1
1

.
.
.
1





; (13)
any solution to (13) satisfies
n

i=1
q
i
= max
(α
1
, ,α
n
)∈R
n

n

i=1
α
i
|A

i
|

2
n

i=1
n

j=1
α
i
α
j
|A
i
∩ A
j
|
. (14)
Proof. Write P =

|A
i
∩A
j
|
|A
i
||A

j
|

n×n
, Q =

|A
i
∩ A
j
|

n×n
and R = diag(1/|A
1
|, . . . , 1/|A
n
|).
Obviously, P = RQR, Q = R
−1
P R
−1
. Let (β
1
, . . . , β
n
) ∈ R
n
be a solution to (12). Then
P






β
1
|A
1
|
β
2
|A
2
|
.
.
.
β
n
|A
n
|





= RR
−1

P R
−1





β
1
β
2
.
.
.
β
n





= RQ





β
1
β

2
.
.
.
β
n





= R





|A
1
|
|A
2
|
.
.
.
|A
n
|






=





1
1
.
.
.
1





.
This solves the existence. Suppose (q
1
, q
2
, · · · , q
n
)
T

is a solution to (13), that is,
RQR





q
1
q
2
.
.
.
q
n





=





1
1
.

.
.
1





⇔ Q





q
1
/|A
1
|
q
2
/|A
2
|
.
.
.
q
n
/|A

n
|





=





|A
1
|
|A
2
|
.
.
.
|A
n
|






.
By (11), (12) and (13),
max
(α
1
, ,α
n
)∈R
n

n

i=1
α
i
|A
i
|

2
n

i=1
n

j=1
α
i
α
j

|A
i
∩ A
j
|
=
n

i=1
n

j=1
q
i
|A
i
|
·
q
j
|A
j
|
· |A
i
∩ A
j
|
= (q
1

, q
2
, · · · , q
n
)P





q
1
q
2
.
.
.
q
n





= (q
1
, q
2
, · · · , q
n

)





1
1
.
.
.
1





=
n

i=1
q
i
.
So we get (14). This concludes the whole proof.
the electronic journal of combinatorics 17 (2010), #R58 5
3.3 A combinatorial proof
In this subsection, we provide a combinatorial proof for the inequality in (11) to help
understand the equality case. To achieve the goal we need only prove




n

i=1
A
i





n

i=1
α
i
|A
i
|

2
n

i=1
n

j=1
α
i

α
j
|A
i
∩ A
j
|
.
holds for all integral weights α
i
∈ Z such that

n
i=1
α
i
|A
i
| > 0. Suppose this is the case.
Let U = ∪
n
i=1
A
i
and χ
i
be the indicator function of A
i
. Define f(x) =


n
i=1
α
i
χ
i
(x) and
for all k ∈ Z,
U
k
.
= {x ∈ U : f(x) = k}, A
k
i
.
= A
i
∩ U
k
.
Obviously, f =

k∈Z
kχ
U
k
. Note
n

i=1

α
i
|A
k
i
| =
n

i=1
α
i

U
χ
A
i
∩U
k =
n

i=1
α
i

U
χ
i
· χ
U
k =


U
f · χ
U
k = k · |U
k
|, (15)
and

k∈Z
k|A
k
i
| =

k∈Z
k

U
χ
i
· χ
U
k =

A
i

k∈Z
kχ

U
k =

A
i
n

j=1
α
j
χ
j
=
n

j=1
α
j
|A
i
∩ A
j
|, (16)
here the integration means

U
g =

x∈U
g(x). By (15),

|U| =

k∈Z
|U
k
| 

k=0

n
i=1
α
i
|X
k
i
|
k
.
Now we need an inequality: for all r, s > 0 one has
1
s

2
r
−
s
r
2


⇔ (
1
s
−
1
r
)
2
 0

.
By (15) again,

n
i=1
α
i
|A
k
i
| and k have the same sign, and consequently for r > 0,

n
i=1
α
i
|A
k
i
|

k


2
r

n
i=1
α
i
|A
k
i
| −
k
r
2

n
i=1
α
i
|A
k
i
| if k > 0
−
2
r


n
i=1
α
i
|A
k
i
| −
k
r
2

n
i=1
α
i
|A
k
i
| if k < 0

2
r
n

i=1
α
i
|A
k

i
| −
k
r
2
n

i=1
α
i
|A
k
i
| if k = 0.
Recall that
2
r

n
i=1
α
i
|A
k
i
| −
k
r
2


n
i=1
α
i
|A
k
i
| = 0 when k = 0. By (16),
|U | 

k∈Z

2
r
n

i=1
α
i
|A
k
i
| −
k
r
2
n

i=1
α

i
|A
k
i
|

=
2
r
n

i=1
α
i
|A
i
| −
1
r
2
n

i=1
n

j=1
α
i
α
j

|A
i
∩ A
j
|
.
= W (r).
the electronic journal of combinatorics 17 (2010), #R58 6
Finally,
|U|  max
r>0
W (r) = W (r
∗
) =

n

i=1
α
i
|A
i
|

2
n

i=1
n


j=1
α
i
α
j
|A
i
∩ A
j
|
,
where r
∗
= (

n
i=1
α
i
|A
i
|)/(

n
i=1

n
j=1
α
i

α
j
|A
i
∩ A
j
|). This concludes the whole proof. A
byproduct of this proof is the following characterization of the equality case:



n

i=1
A
i



=

n

i=1
α
i
|A
i
|


2
n

i=1
n

j=1
α
i
α
j
|A
i
∩ A
j
|
⇔
n

i=1
α
i
χ
i
(x)



S
n

i=1
A
i
is a non-zero constant function.
4 From Functional Analysis to Probability Theory
4.1 Finitely many events
In this section we choose H to be the L
2
space of the given probability space (Ω, F, P ). Let
E, F be two events and χ
E
, χ
F
be the corresponding indicator functions. It is well-known
that Hilbert space theory and probability theory are intimately connected by
< χ
E
, χ
F
>= P (E ∩ F ).
Note χ
E
, χ
F
are orthogo na l means E, F are disjoint. G iven events {E
i
}
n
i=1
, define y

i
=
χ
E
i
(i ∈ [n]) and x = χ
∪
i
E
i
. By (2) and (3), we extend the Gallot -Kounias inequality
[9, 11] to its full generality in the f ollowing form.
Theorem 4.1 (Gallot-Kounias).
P (
n

i=1
E
i
)  max
(α
1
, ,α
n
)∈R
n

n

i=1

α
i
P (E
i
)

2
n

i=1
n

j=1
α
i
α
j
P (E
i
∩ E
j
)
=
n

i=1
n

j=1
γ

i
γ
j
P (E
i
∩ E
j
), (17)
where (γ
1
, . . . , γ
n
) ∈ R
n
is any solution to





P (E
1
∩ E
1
) P (E
1
∩ E
2
) · · · P (E
1

∩ E
n
)
P (E
2
∩ E
1
) P (E
2
∩ E
2
) · · · P (E
2
∩ E
n
)
.
.
.
.
.
.
.
.
.
.
.
.
P (E
n

∩ E
1
) P (E
n
∩ E
2
) · · · P (E
n
∩ E
n
)










γ
1
γ
2
.
.
.
γ
n






=





P (E
1
)
P (E
2
)
.
.
.
P (E
n
)





. (18)
the electronic journal of combinatorics 17 (2010), #R58 7

To the authors’ knowledge, it seems that the Gallot-Kounias inequality, being discov-
ered 40 years ago, was almost forg otten by Mathematicians. Gallot and Kounias originally
expressed their results in terms of generalized inverse of matrices, and this may prevent
their results from being appreciated by others. So we restate their results in a more natural
way in Theorem 4.1. Note in this context (10) turns out to be the original Chung-Erd˝os
inequality [3]
P (
n

i=1
E
i
) 

n

i=1
P (E
i
)

2
n

i=1
n

j=1
P (E
i

∩ E
j
)
, (19)
and the Bessel inequality (9) turns out to be a trivial equality. Also note that
sup
α
i
>0

n

i=1
α
i
P (E
i
)

2
n

i=1
n

j=1
α
i
α
j

P (E
i
∩ E
j
)
= sup
α
i
>0

n

i=1
α
i
P (E
i
)

2
n

i=1
n

j=1
α
2
i
P (E

i
∩ E
j
)
= sup
α
i
>0
n

i=1
α
i
P (E
i
)
2
n

j=1
α
j
P (E
i
∩ E
j
)
.
Similar to Theorem 3.2 one can establish the following theorem.
Theorem 4.2. The following matrix equation always has a solution


P (E
i
∩ E
j
)
P (E
i
)P (E
j
)

n×n





q
1
q
2
.
.
.
q
n






=





1
1
.
.
.
1





; (20)
any solution to (20) satisfies
n

i=1
q
i
= max
(α
1
, ,α

n
)∈R
n

n

i=1
α
i
P (E
i
)

2
n

i=1
n

j=1
α
i
α
j
P (E
i
∩ E
j
)
. (21)

4.2 Borel-Cantelli lemma
Let {E
i
}
∞
i=1
be infinitely many events on the probability space (Ω, F, P ). The Borel-
Cantelli lemma states that: (a) if

∞
i=1
P (E
i
) < ∞, then P (lim sup E
i
) = 0; (b) if

∞
i=1
P (E
i
) = ∞ and {E
i
}
∞
i=1
are mutually independent, then P (lim sup E
i
) = 1. Here
lim sup E

i
= ∩
∞
i=1
∪
∞
k=i
E
k
. The Borel-Cantelli lemma played an exceptionally important
role in probability theory, and many investigations were devoted to the second part of the
Borel-Cantelli lemma in the attempt to weaken the independence condition on {E
i
}
∞
i=1
.
the electronic journal of combinatorics 17 (2010), #R58 8
Towards this question, Erd˝os and R´enyi [6, 14 ] obtained a nice result closely related to
(19): if

∞
i=1
P (E
i
) = ∞, then
P (lim sup E
i
)  lim sup
n→∞


n

k=1
P (E
k
)

2
n

i=1
n

j=1
P (E
i
∩ E
j
)
. (22)
Recently, by carefully studying the effect of the denominator in the right hand of (2 2),
the authors [8] established a weighted version of the Erd˝os-R´enyi theorem which states:
Theorem 4.3 (Feng-Li-Shen). If

∞
i=1
α
i
P (E

i
) = ∞, then
P (lim sup E
i
)  lim sup
n→∞

n

k=1
α
k
P (E
k
)

2
n

i=1
n

j=1
α
i
α
j
P (E
i
∩ E

j
)
. (23)
5 Applications of the de Caen-Selbe r g Inequality
5.1 The finite field Kakeya set
Let F
q
denote a finite field of q elements. Define a set K ⊂ F
n
q
to be Kakeya if it contains
a translate of any given line. The finite field Kakeya problem, posed by Wolff in his
influential survey [17 ], conjectured that |K|  C
n
q
n
holds for some constant C
n
. Recently,
using the polynomial method in algebraic extremal combinatorics, D vir [4] completely
confirmed this conjecture by proving
|K| 

n + q − 1
n

. (24)
If n = 2, it is well-known that (24) is sharp [7] and can be established by a simple counting
argument [15]. For n  3 , see [16] for further improvement.
Similarly, we say a subset E ⊂ F

n
q
is an (n, k)-set if it contains a translate of any given
k-plane. Ellenberg, Oberlin and Tao [5] proved that if 2  k < n, t hen
|E|  q
n
−

n
2

q
n−k+1
+ o(q
n−k+1
) (q → ∞). (25)
Using the de Caen-Selberg inequality we can slightly improve (25) when k = n − 1  2.
Theorem 5.1. Any (n, n − 1)-s e t E ⊂ F
n
q
(n  3) satisfies
|E|  q
n
− q
2
+ o(q
2
) (q → ∞),
where F
q

denotes a finite field of q elements.
the electronic journal of combinatorics 17 (2010), #R58 9
Proof. Since the total number s of (n − 1)-dimensional hyperplanes passing through the
origin equals the total number of lines passing through the origin,
s =
q
n
− 1
q − 1
.
Let {P
i
}
s
i=1
be such hyperplanes. By the de Caen-Selberg inequality (4),
|E| 
s

i=1
|P
i
|
2
s

j=1
|P
i
∩ P

j
|

s · q
2n−2
q
n−1
+ (s − 1)q
n−2
=
s · q
2n−2
+ q
n
(q
n−1
− q
n−2
) − q
n
(q
n−1
− q
n−2
)
(q
n−1
− q
n−2
) + s · q

n−2
= q
n
−
q
n
(q
n−1
− q
n−2
)
q
n−1
+ (s − 1)q
n−2
= q
n
− q
2
+ o(q
2
) (q → ∞).
5.2 The finite field Nikodym set
Define a set B ⊂ F
n
q
to be Nikodym if for each z ∈ B
c
there exists a line L
z

passing
through z such that L
z
\{z} ⊂ B. Obviously, all such lines {L
z
}
z∈B
c
are different from
each other. Similar to (24) Li [12] proved (i)
|B| 

n + q − 2
n

; (26)
(ii) any two-dimensional Nikodym set B ⊂ F
2
q
satisfies
|B| 
2q
2
3
+ O(q) (q → ∞). (27)
Using the de Caen-Selberg inequality we can improve (27) substantially as follows, which
shows some difference between the two-dimensional Kakeya sets and Nikodym sets.
Theorem 5.2. Any Nikodym set B ⊂ F
2
q

satisfies
|B|  q
2
− q
3/2
− q,
where F
q
denotes a finite field of q elements.
Proof. L et s = |B
c
|. By the de Caen-Selberg inequality (4),
q
2
− s = |B| 




z∈B
c
L
z
\{z}




s


i=1
(q − 1)
2
(q − 1) + s − 1
=
s(q − 1)
2
s + q − 2
.
the electronic journal of combinatorics 17 (2010), #R58 10
Equivalently,
s
2
− (q + 1)s − q
2
(q − 2)  0.
Hence
|B| = q
2
− s  q
2
−
q + 1 +

(q + 1)
2
+ 4q
2
(q − 2)
2

 q
2
− q
3/2
− q.
We thank a referee for many valuable suggestions leading to the clear presentat io n of
the paper.
References
[1] E. Bombieri, A note on the large sieve. Acta Arith. 18 (1971) 401–404.
[2] D. de Caen, A lower bound on the probability of a union. Discrete Math. 169 (1997)
217–220.
[3] K. L. Chung, P. Erd˝os, On the application of the Bore l - Cantelli lemma. Trans. Amer.
Math. Soc. 72 (1952) 179–186.
[4] Z. Dvir, On the size of Kakeya sets in finite fields. To appear in J. Amer. Math. Soc.
[5] J. S. Ellenberg, R. Oberlin, T. Tao, The Ka k eya set and maximal conjectures f o r
algebraic varieties over finite fields. Preprint.
[6] P. Erd˝os, A. R´enyi, O n Cantor’s series with convergent

1/q
n
. Ann. Univ. Sci.
Budapest. E˝otv˝os Sect. Math. 2 (1959) 93–109.
[7] X. W. C. Faber, On the finite field Kakeya problem in two dimensions, J. Number
Theory 124 (2007) 248–257.
[8] C. Feng, L. L i, J. Shen, On the Borel-Cantelli lemma and its generalization. Comptes
Rendus Mathematique 347 (2009) 1313–1316.
[9] S. Gallot , A bound for the maximum of a number of random variables. J. Appl. Prob.
3 (1966) 556–558.
[10] H. Heilbronn, On the averages of some arithmetical functions of two variabl es. Math-
ematika 5 (1958) 1–7.

[11] E. G. Kounias, Bounds for the probability of a union, with app l i cations. Ann. Math.
Statist. 39 (1968) 2154–2158.
[12] L. Li, On the size of Nikodym sets in finite fields. Preprint.
[13] J. E. Pe˘cari´v, On some classical inequalities in unitary spaces. Mat. Bilten 42 (1992)
63–72.
[14] A. R´enyi, Probability Theory. North-Holland Series in Applied Mathematics and Me-
chanics, Vol. 10. North-Holland Publishing Co., Amsterdam-London, 1970; German
version 1962, French version 1966, new Hungarian edition 1965.
the electronic journal of combinatorics 17 (2010), #R58 11
[15] K. M. Rogers, The finite field Kakeya problem, Amer. Math. Monthly 108 (2000)
756–759.
[16] S. Saraf, M. Sudan, Improved lower bound on the size of Ka keya se ts over finite fields .
Preprint.
[17] T. Wolff, Recent work connectecd with the Kakeya problem. Prospects in Mathematics
(Princeton, NJ, 1996), Amer. Math. Soc. (1 999) 129–162.
the electronic journal of combinatorics 17 (2010), #R58 12