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: fcr@sjtu.edu.cn. 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- iangpan@yahoo.com.cn. 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: js48@txstate.edu. 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 [...]... a number of random variables J Appl Prob 3 (1966) 556–558 [10] H Heilbronn, On the averages of some arithmetical functions of two variables Mathematika 5 (1958) 1–7 [11] E G Kounias, Bounds for the probability of a union, with applications Ann Math Statist 39 (1968) 2154–2158 [12] L Li, On the size of Nikodym sets in finite fields Preprint [13] J E Pe˘ari´, On some classical inequalities in unitary spaces... classical inequalities in unitary spaces Mat Bilten 42 (1992) c v 63–72 [14] A R´nyi, Probability Theory North-Holland Series in Applied Mathematics and Mee 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... 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