ON THE SIGNED SMALL BALL INEQUALITY WITH
RESTRICTED COEFFICIENTS
DIMITRIOS KARSLIDIS
Abstract. Let R = R1 × R2 × · · · × Rd denote a dyadic rectangle in the unit
cube [0, 1]d , d ≥ 3. Let hR be the L∞ normalized Haar function supported
on R. We show that for all integers n ≥ 1, the conjectured signed small ball
inequality,
X
d
αR hR & n 2
where αR ∈ {±1},
−n
|R|=2
∞
holds under the additional assumption that the coefficients αR also satisfy the
“splitting property”, αR = αR1 · αR2 ×R3 ×···×Rd with αR1 , αR2 ×R3 ×···×Rd ∈
{±1}. The unrestricted small ball inequality has connections to multiple fields,
such as probability, approximation and discrepancy.
Subject Class: [2010] Primary: 26B15; 42B25; Secondary: 60G50; 60E15.
Keywords: small ball inequality, Littlewood-Paley Inequalities, Haar functions,
dyadic expansion, binary random variables.
1. Introduction
The small ball inequality (described in Section 1.3) is a statement concerning
the sup norm of a finite linear combination of Haar functions. The sharp constant
in this inequality, as yet unproven, is of considerable interest due to a variety of
applications in probability and discrepancy theory. In this paper, we prove the
sharp version of the signed small ball inequality under a structural assumption on
the coefficients accompanying the Haar functions.
1.1. Preliminaries. We begin by defining the concepts needed to state the general
conjectured small ball inequality.
Let 1I (x) be the characteristic function of the interval I, i.e.
(
1, x ∈ I
1I (x) =
0, otherwise.
Consider the collection of the dyadic intervals of [0, 1]:
m m+1
n
D = I = n,
: m, n ∈ Z, n ≥ 0, 0 ≤ m < 2
with
2
2n
(1)
D∗ = D ∪ {[−1, 1]}.
email: dimkars@math.ubc.ca
The author was partially supported by an NSERC Discovery Grant and Idryma Paideias kai
Eyropaikoy Politismoy.
1
2
DIMITRIOS KARSLIDIS
If we consider two distinct dyadic intervals, then either one will be strictly contained in the other, or they will be disjoint. Moreover, for every interval I ∈ D, its
left and right halves (denoted by Il and Ir respectively) are also dyadic. We define
the L∞ normalized Haar function, hI , corresponding to an interval I as:
hI (x) = − 1Il + 1Ir .
(2)
Haar functions can be easily extended to higher dimensions. In order to do so, we
consider the family of dyadic rectangles in dimension d ≥ 2:
Dd = {R = R1 × . . . × Rd : Rj ∈ D},
i.e. every R ∈ Dd is a Cartesian product of dyadic intervals. The Haar functions
supported on R are defined as a coordinate-wise product of one-dimensional Haar
functions:
(3)
hR (x1 , . . . , xd ) = hI1 (x1 ) · · · hId (xd ), where R = I1 × . . . × Id , Ij ∈ D.
Haar functions enjoy the following properties:
h2R (x) = 1R (x),
(4)
Z
(5)
hR0 (x)hR00 (x)dx = 0,
[0,1]d
whenever R0 6= R00 ,
Z
hR (x)dx = 0, ∀ R ∈ Dd ,
(6)
[0,1]d
and the collection H = {hR }R∈D∗d forms an unconditional basis for the Lebesgue
spaces, Lp , 1 < p < ∞.
Let
d
Hdn = {~r ∈ Z+
: |~r| := r1 + r2 · · · +rd = n}
with
Zd+ = {~r = (r1 , . . . , rd ) : rj ≥ 0 and rj ∈ Z, j = 1, . . . , d},
and
R~r = {R ∈ Dd : Ri ∈ D and |Rj | = 2−rj , j = 1, . . . , d}.
In other words, the set R~r consists of all the dyadic rectangles that have the same
shape. The rectangles in R~r are disjoint and partition the d-dimensional unit cube,
[0, 1)d . A function defined on [0, 1]d of the form
X
f~r =
R hR with R = ±1
R∈R~
r
is called an ~r− function with parameter ~r ∈ Hdn . These functions are also known in
the literature as generalized Rademacher functions. It can easily be verified that
~r− functions are orthonormal with respect to the L2 norm.
1.2. Acknowledgements. I would like to thank my advisor Malabika Pramanik
for introducing me to this problem and the numerous discussions that led to this
paper. I am grateful to the anonymous referee for valuable comments concerning
the content of this article. I am also grateful to Professors Richard Froese and
Philip Loewen for their input following a careful reading of the manuscript.
ON THE SIGNED SMALL BALL INEQUALITY WITH RESTRICTED COEFFICIENTS
3
1.3. Statement of the conjectured Small Ball Inequality. For brevity, let
Adn = {R ∈ Dd : |R| = 2−n }, i.e. the set of all dyadic rectangles whose ddimensional volume is equal to 2−n . Moreover, A & B means that there is a
constant K such that A ≥ KB. In our setting, K does not depend on n or {αR }.
Conjecture 1.1. (The Small Ball Conjecture):
In all dimensions d ≥ 2, for any choice of the coefficients αR , and all integers
n ≥ 1, one has the following inequality
X
d−2 X
αR hR ≥ Cd · 2−n
|αR |,
(7)
n 2 d
d
R∈An
∞
R∈An
where Cd is a constant that depends only on the dimension d but not on n or the
choice of {aR }.
The critical feature of this inequality is the precise exponent of n occurring on the
d−2
d−1
left side. If we replace n 2 with n 2 , then we get the so-called “trivial bound”:
(8)
n
d−1
2
X
αR hR d
R∈An
& 2−n
X
|αR |.
R∈Ad
n
L2
If we compare the conjectured small ball inequality with the inequality (8), but with
the L2 norm replaced by the L∞ norm, then we√can see that the conjectured small
ball inequality is better than (8) by a factor of n. The proof of the trivial bound
(8) can be found in [7]. Also, if we choose each αR in the collection of independent
random variables such that αR = ±1, then one verifies that this conjecture is sharp
(see [3, 20]).
1.4. Recent History. The small ball conjecture has been proved in d = 2 by
M. Talagrand [17] in 1994. In 1995, V. Temlyakov [19] gave another proof of this
inequality. The first improvement over the trivial bound in higher dimensions,
specifically in dimension d = 3, by a factor logarithmic in n, was obtained by
J. Beck [2]. In 2008, a body of work authored by D. Bilyk, M. Lacey, and A.
Vagharshakyan [3], [4] made significant progress toward the study of the structure
of the small ball inequality. They proved the following theorem:
Theorem 1.2. In all dimensions d ≥ 3, there exists 12 > η(d) > 0 such that for all
choices of coefficients αR and all non-negative integers n we have the inequality
X
X
d−1
−η(d) 2
(9)
n
αR hR & 2−n
|αR |.
d
d
R∈An
∞
R∈An
The results of [3, 4] provide unspecified small values of η(d), whereas the small
ball conjecture says that (9) should hold with η = 12 .
A complete resolution of the small ball conjecture appears to be a difficult
problem, but the following special case, while still unsolved, appears to be more
tractable.
4
DIMITRIOS KARSLIDIS
Conjecture 1.3. (The signed small ball conjecture) If αR = ±1 for every R ∈ Adn ,
then we have
X
d
αR hR & n 2 .
(10)
d
∞
R∈An
If
Hn =
X
αR hR =
R∈Ad
n
X
f~r ,
~
r ∈Hd
n
then replacing the exponent of the integer n in the above conjecture by
us the trivial bound on Hn , which is
X d−1
2 ,
(11)
f~r &n
~r∈Hd 2
n
d−1
2
gives
L
which can be obtained, using the orthogonality of generalized Rademacher functions
and the fact that #Hdn , the cardinality of the set Hdn is of order nd−1 . The exponent
of the integer n appears naturally in the trivial bound (11). This can be explained
by observing that “the volume constraint”, |R| = 2−n , on dyadic rectangles, or
equivalently |~r| = n, reduces the number of “free” parameters in the vector ~r ∈ Hdn
by one. Therefore, the total number of terms in the sum is of order nd−1 while
the conjectured signed small
√ ball inequality requires the “frozen” parameter to
contribute by a factor of n rather than by a factor of 1 as in the trivial estimate
(11). In [5], D. Bilyk, M. Lacey, and A. Vagharshakyan quantified explicitly the
improvement over the trivial bound, i.e. they proved the signed version of the
Theorem 1.2, explicitly providing the range for the values of η. In particular, they
showed
X
d−1
1
(12)
αR hR ≥ Cd · n 2 + 8d − , where αR ∈ {±1},
d
R∈An
∞
for every > 0. Under an additional assumption on the length of the first side of
a dyadic rectangle R, in [6], D. Bilyk, M. Lacey, I. Parissis and A. Vagharshakyan
managed to get a better exponent of the integer n than the one given in the relation
(12) when d = 3. Specifically, they proved
X
9
(13)
αR hR & n 8 , where αR ∈ {±1}.
−n
|R|=2
|R1 |≥2−n/2
∞
1.5. Background. Sup norms bounds identical or similar to (7) have appeared
in probability, approximation and discrepancy theory. We briefly outline some of
these connections here, and refer the interested reader to the excellent survey [7]
by D. Bilyk for further details.
1.5.1. Brownian Sheet asymptotics. The name “small ball inequality” has its roots
in probability theory, where the question of interest there is determining the exact
asymptotic behaviour of the small deviation probability, φ(ε) := − log P(kBkL∞ ([0,1]d ) <
ON THE SIGNED SMALL BALL INEQUALITY WITH RESTRICTED COEFFICIENTS
5
ε as ε → 0+ . Here, B is the Brownian sheet, i.e. a centered multiparameter Gaussian process characterized by the covariance relation
(14)
E(B(s)B(t)) =
d
Y
min(sj , tj ),
j=1
for s, t ∈ [0, 1]d .
Conjecture 1.4. In dimension d ≥ 2, for the Brownian sheet B we have
2d−1
2d−1
1
1
1
0 1
≤ φ(ε) ≤ C 2 log
,
(15)
C 2 log
ε
ε
ε
ε
for small ε > 0, where C and C 0 are just absolute constants depending only on the
dimension d.
The above double-sided inequality is usually expressed in a compact way as
2d−1
φ(ε) ' ε12 log 1ε
.
The upper bound in (15) was proven by Bass (d = 2), and by T. Dunker, T.
Kühn, M. Lifshits and W. Linde (d ≥ 3) in [1, 9]. In d = 2, the conjecture has been
solved by Talagrand [17]. The trivial lower bound,
2d−2
1
1
(16)
C 0 2 log
≤ φ(ε) for small ε > 0,
ε
ε
was shown in [8]. Based on their work on the small ball inequality [3, 4], D. Bilyk,
M. Lacey, and A. Vagharshakyan obtained a gain over the trivial bound (16) by
improving the exponent to 2d − 2 + θ, where θ = θ(d) > 0 is a small parameter.
Moreover, it has been shown that if a continuous version of the conjectured small
ball inequality holds with the Haar functions replaced by continuous wavelets, then
the lower bound in (15) would hold (see [7]).
1.5.2. Covering Numbers. Let B∞ and B(Lp ) denote the unit ball on L∞ ([0, 1]d )
and Lp ([0, 1]d ) respectively, and B(M W p ) := Td (B(Lp )), where Td is the integral
operator defined as
Td : Lp ([0, 1]d ) → C([0, 1]d ), with
Z x1
Z xd
(17)
(Td f )(x1 , . . . , xd ) :=
···
f (y1 , . . . , yd )dy1 . . . dyd .
0
0
Experts in approximation theory are interested in determining the asymptotic behaviour of ψ(ε) := log(N (ε, p, d)) as ε → 0+ , where
(
)
N
[
N
p
p
N (ε, p, d) := min N : ∃{xk }k=1 ⊂ B(M W ) such that B(M W ) ⊂
xk + εB∞
k=1
is a covering number which expresses the smallest number of balls of radius ε needed
to cover the unit ball B(M W p ). Kuelbs and Li have established a deep connection
between the asymptotic behaviour of φ(ε) and ψ(ε) [10]. Their results imply, in
particular, the following theorem:
Theorem 1.5. In d ≥ 2, for small ε > 0 we have
β
β
1 2
1
−1
−2
if and only if ψ(ε) ' ε
log
.
φ(ε) ' ε
log
ε
ε
6
DIMITRIOS KARSLIDIS
Therefore, the equivalent conjecture in approximation theory is
Conjecture 1.6. In d ≥ 2, for small ε > 0 we have that
1 2d−1
ψ(ε) ' ε−1 (log ) 2 .
ε
All the results obtained regarding small deviation probability can be adapted to
the case of covering numbers with the aid of Theorem 1.5. In addition, using Theorem 1.5, it can be deduced that the conjectured small ball inequality implies the
lower bound in Conjecture 1.6, although the lower bound on ψ(ε) can be obtained
directly from the conjectured small ball inequality [7].
1.5.3. Geometric Discrepancy. One of the most challenging questions in geometric
discrepancy theory is the following: given a set PN ⊂ [0, 1]d with cardinality #PN =
N , how small can the discrepancy function,
(18)
DN (x1 , . . . , xd ) = #{PN ∩ [0, x1 ) × · · · × [0, xd )} − N x1 · · · xd ,
be in some norm? Relation (18) is the difference between the actual and expected
number of points in the box [0, x1 ) × · · · × [0, xd ). Usually the norm of interest is
the L∞ norm since it measures how well distributed the set PN is.
Conjecture 1.7. For all d ≥ 2 and all PN ⊂ [0, 1]d with #PN = N , we have the
estimate
d
(19)
kDN k∞ ≥ Cd (log N ) 2 .
The first quantitative bound of the function DN (·) was obtained by K. F. Roth
[12, 1954]. He showed that for all d ≥ 2,
(20)
kDN k2 ≥ Cd (log N )
d−1
2
,
d−1
2
(N ). It is worth noticing that the exponent of
which implies kDN k∞ ≥ Cd log
the integer n appearing in the trivial estimate (11) and Conjecture 1.3 is exactly
the same as in the trivial bound (20) and Conjecture 1.7 respectively. In d = 2,
Conjecture 1.7 was proven by W. M. Schmidt [13, 1972]. Much later, only partial
results were established in higher dimensions. In d = 3, J. Beck [2, 1989] showed
that
1
(21)
kDN k∞ ≥ C log N · (log log N ) 8 −ε ,
1
i.e. the gain over the trivial estimate (20) was of order (log log N ) 8 −ε . In 2008, D.
Bilyk, M. Lacey and A. Vagharshakyan [3, 4] made the first improvement in (21).
More precisely, they proved the following theorem:
Theorem 1.8. For all d ≥ 3, there exists some 0 < η = η(d) <
PN ⊂ [0, 1]d with #PN = N , we have the estimate
(22)
kDN k∞ ≥ Cd (log N )
1
2
such that for all
d−1
2 +η .
Starting with Roth’s work, most progress on this problem relied on the following
approach: in Haar basis expansion of the discrepancy function DN , most of the
information is carried by the terms corresponding to rectangles of size |R| ≈ N1 . In
particular, if 2N ≤ 2n < 4N , then for each ~r ∈ Hdn one can choose signs εR = ±1
so that hDN , f~r i & 1. It is therefore not surprising that the L2 estimate (20) and
the L∞ conjecture (19) are almost identical to the corresponding bounds (11) and
ON THE SIGNED SMALL BALL INEQUALITY WITH RESTRICTED COEFFICIENTS
7
(10) in the case of the signed small ball inequality.
Currently, there is no indication that the conjectured small ball inequality implies
inequality (19), but a possible connection could come from the method of proof.
That is, Roth’s Haar function approach allows one to adapt the progress on the
small ball conjecture to give partial results towards inequality (19).
1.6. Statement of the main result. In this paper, we prove a restricted version
of the conjectured signed small ball inequality obtaining the sharp exponent under
an additional constraint on the coefficients {αR : R ∈ Dd , |R| = 2−n }. More
precisely,
Theorem 1.9. Let hR denote an L∞ normalized Haar function supported on dyadic
rectangle R = R1 × R0 ∈ Dd with R1 ∈ D and R0 = R2 × · · · ×Rd ∈ Dd−1 where d ≥
3. Then for all integers n ≥ 1 and all choices of coefficients (αR )R∈Dd ⊂ {−1, 1}
which satisfy the “splitting property”, αR = αR1 αR0 with αR1 , αR0 ∈ {−1, 1}, we
have the inequality
X
d
(23)
αR hR & n 2 .
d
R∈An
∞
2. Auxiliary Results
In this section, we record some notation and auxiliary results which will be used
in the proof of the main theorem.
We denote the function inside the L∞ norm, k · kL∞ , in (23) by
(24)
Hn (~x) =
X
αR hR (~x) =
n
X
X
αR1 hR1 (x1 )Fr1 (~x0 ),
r1 =0 R∈A1r
R∈Ad
n
1
where
(25)
Fr1 (~x0 ) =
X
αR0 hR0 (~x0 ) with ~x = (x1 , ~x0 ) ∈ [0, 1)d , and
R0 ∈Ad−1
n−r1
~x0 = (x2 , x3 , . . . , xd ) ∈ [0, 1)d−1 .
The first auxiliary results of this section are Littlewood-Paley inequalities. These
will be very valuable in proving Proposition 2.1 and the proof of Littlewood-Paley
inequalities can be found in [7].
To each function of the form
X
(26)
f (~x) =
R hR (~x), where ~x ∈ [0, 1]d ,
R∈D∗d
we associate the expression

(27)
(Sd f )(~x) = 
 21
X
|R |2 1R (~x) , with (R )R∈D∗d ⊂ R,
R∈D∗d
which is called the product dyadic square function of f . The product LittlewoodPaley inequalities state that
(Ap )d kSd kp ≤ kf kp ≤ (Bp )d kSd kp , for p ∈ (1, ∞).
8
DIMITRIOS KARSLIDIS
The next proposition will help us to compare the quantity kf kp for different values of
p in (0, ∞). In particular, when f is a special linear combination of Haar functions,
the next proposition shows that the norms kf kp for all p are comparable. This
proposition should be viewed as a generalization of Khintchine’s inequality. The
proof follows the strategy outlined in Corollary 1.4 stated in [18, page 5-6], and we
extend that Corollary to dimensions greater than one. Specifically, we establish the
following:
Proposition 2.1. Let f be a linear combination of Haar functions, i.e.
X
f (~x) =
R hR (~x),
R∈Ad
n
such that the square function, (Sd f )(·), is a constant on [0, 1]d (i.e. (Sd f )(~x) =
c(n, d) for every ~x ∈ [0, 1]d , where the constant c(n, d) depends on the integer
n ≥ 1 and on the dimension d). Then there exist positive constants c1 (p, q, d)
and c2 (p, q, d) such that
(28)
c2 kf kq ≤ kf kp ≤ c1 kf kq
for every
0 < p < q < ∞.
Proof. First, we observe that the right side of inequality (28) is a simple consequence
of Hölder’s inequality (c1 = 1). Therefore, our goal is to prove that
c2 kf kq ≤ kf kp
for every
0 < p < q < ∞.
We will show the above inequality by considering the following three cases:
Case 1: 1 < p < q < ∞
Case 2: 0 < p ≤ 1 < q < ∞
Case 3: 0 < p < q ≤ 1.
In Case 1, using the Littlewood-Paley Inequality and the fact that (Sd f )(·) is independent of ~x, we get that
d
Ap
kf kq .
(29)
kf kp ≥ (Ap )d kSd f kp = (Ap )d kSd f kq ≥
Bq
d
A
Thus, c2 kf kq ≤ kf kp with c2 = Bpq .
Now, we turn our attention to the proof of Case 2. First, choose α ∈ (0, 1) such
p
r
that p > αq. Set r = αq
and s = r−1
(i.e. 1r + 1s = 1). Also set b = 1 − α so
that α + b = 1. Note that αr < 1 < bs. Indeed, αr = pq < 1, and bs > 1 ⇔
r
(1 − α) r−1
> 1 ⇔ (1 − α)r > r − 1 ⇔ 1 > αr but this is true as it was shown
earlier. Now, applying Hölder’s inequality, we get that
Z
bq
(30)
kf kqq =
|f (x)|αq |f (x)|bq dx ≤ kf kαq
αrq kf kbsq .
[0,1]d
But bs > 1 ⇒ bsq > q > 1 and Case 1 implies that there exists a positive constant
c2 such that
(31)
bq
(c2 )bq kf kbq
bsq ≤ kf kq .
Combining inequalities (30) and (31), we get that
b
bq
q(1−b)
−α
kf kqq ≤ (c2 )−bq kf kαq
≤ (c2 )−bq kf kαq
kf kαrq ,
αrq kf kq ⇒ kf kq
αrq ⇒ kf kq ≤ (c2 )
and recall that αrq = p.
ON THE SIGNED SMALL BALL INEQUALITY WITH RESTRICTED COEFFICIENTS
9
Finally, we prove Case 3. Define the numbers α, b, r and s in the same way as in
Case 2 and note that inequality (30) still holds. Since q ≤ 1 ⇒ bsq ≤ bs, and using
the right side of inequality (28), we get
bq
kf kbq
bsq ≤ kf kbs .
(32)
Now, putting together inequalities (30) and (32), we get that
bq
kf kqq ≤ kf kαq
αrq kf kbs .
(33)
−bq
Since bs > 1 ≥ q, Case 2 gives us kf kbq
kf kbq
q . Combining the previous
bs ≤ (c2 )
inequality with inequality (33), we obtain
b
q(1−b)
bq
−a
kf karq ,
kf kqq ≤ (c2 )−bq kf kαq
≤ (c2 )−bq kf kαq
αrq kf kq ⇒ kf kq
arq ⇒ kf kq ≤ (c2 )
with αrq = p and this completes the proof of Case 3.
Remark 2.2. It is clear that if
f (~x) =
X
R hR (~x),
R∈Ad
n
with (R )R∈Dd :|R|=2−n ⊂ {−1, 1}, then the square function, (Sd f )(·), is independent
of ~x ∈ [0, 1]d . More precisely,
"
# 12
X
1
2
(Sd f )(~x) =
|R | 1R (~x) = #Hdn 2 ,
R∈Ad
n
where #Hdn denotes the cardinality of the set Hdn . An immediate consequence of
this is that
(34)
kFr1 kL1 ([0,1]d−1 ) & kFr1 kL2 ([0,1]d−1 ) & (n − r1 )
d−2
2
,
r1 = 0, 1, . . . , n,
where Fr1 (·) is as in (25).
The above estimate will be one of the main components that contribute to the
sharp exponent of the integer n ≥ 1 in the conjectured signed small ball inequality.
The last result of this section gives us an explicit formula for the L∞ norm of a
linear combination of iid ±1 random variables which will be useful in the proof of
the Theorem 1.9.
Proposition 2.3. If (Ω, F, P ) is a probability space and (Ui (·))ni=1 is a finite
collection of independent random variables with Ui (·) : Ω → {−1, 1} such that
P (Ui = ±1) > 0 for every i = 1, . . . , n and (ci )ni=1 is any finite sequence of real
numbers, then
n
n
X
X
ci U i =
|ci |.
i=1
∞
i=1
Pn
Pn
Proof. We can see that | i=1 ci Ui (ω)| ≤ i=1 |ci | for every ω ∈ Ω. In order to
complete the proof, we have to find ω0 ∈ Ω such that Ui (ω0 ) = sgn(ci ), i =
1, . . . , n, where
(
1
if x > 0
sgn(x) =
−1 if x ≤ 0.
10
DIMITRIOS KARSLIDIS
Tn
It suffices to show that the set A = i=1 {ω ∈ Ω : Ui (ω) = sgn(ci )} is not a set
n
of measure
Qnzero. Using the independence of the random variables (Ui )i=1 , we get
P (A) = i=1 P (Ui = sgn(ci )) > 0 and which shows that A is not a set of measure
zero.
3. Representation of H in terms of binary random variables
At this point, our goal is to represent the function
X
Hn =
f~r
|~
r |=n
in terms of binary random variables. It will suffice to represent the ~r- functions in
terms of iid ±1 valued random variables for every ~r ∈ Hdn . In particular, taking
into account the ”splitting property” satisfied by the coefficients {αR } and writing
X
f~r (~x) =
αR1 hR1 (x1 )f~r0 (~x0 ),
|R1 |=2−r1
d−1
, we are interested in exwhere ~r = (r1 , ~r0 ) ∈ Hdn with ~r0 = (r2 , . . . , rd ) ∈ Hn−r
1
pressing only one part of an ~r− function, f~r (·), in terms of binary random variables.
More specifically, we are interested in expressing the part which contains the linear
combination of Haar functions in the first coordinate, although the same process
can be applied to other coordinates as well.
It is well known that every point x ∈ [0, 1) can be written, using its dyadic
expansion as:
∞
X
di (x)
(35)
x=
, where the sequence {di (x)}i≥1 ⊂ {0, 1}.
2i
i=1
We also agree to use the terminating dyadic expansion for the dyadic rationals
{x = 2km : k, m ∈ Z, m ≥ 0, 0 ≤ k < 2m }. Next, we gather some useful facts about
the dyadic sequence, {di (x)}i≥1 , which will be used throughout the paper. (they
can be found in [11, page 99-100]) Consider the following probability space
(Ω, B, P ) = ([0, 1), B([0, 1)), | · |),
where |·| is Lebesgue measure and B([0, 1)) is Borel σ -algebra on the interval [0, 1).
Fact 1: Each di is a random variable.
Fact 2: The sequence {di , i ≥ 1} is iid. In other words, {di } is identically
distributed with P [di = 1] = P [di = 0] = 12 and it is independent.
To each such dyadic sequence {di }i≥1 , we associate a sequence of iid ±1 valued
random variables {Xi }i≥1 defined by
(
−1 if di = 0
Xi =
1
if di = 1,
where i = 1, 2, . . .
For every i ∈ Z+ , we denote by
~ i = (X1 , . . . , Xi ),
X
ON THE SIGNED SMALL BALL INEQUALITY WITH RESTRICTED COEFFICIENTS
11
d~i = (d1 , . . . , di ),
~bi = (b1 , b2 , . . . , bi ) ∈ {−1, 1}i ,
~0 = 0 and ~b0 = 0.
with the convention {−1, 1}0 = {0}, d~0 = 0, X
Now, we introduce a map which will be an important ingredient in representing ~r−
functions in terms of of iid random variables {Xi }i≥1. Fix k ∈ {0, 1, . . . , n} and a
collection of numbers {αI }I∈D:|I|=2−k ⊂ {−1, 1}, define the following map:
ak : {−1, 1}k → {αI }I∈D:|I|=2−k ,
(36)
such that for every ~bk ∈ {−1, 1}k ,
ak (~bk ) = αI ,
where I is the dyadic interval with |I| = 2−k given by
" k
!
k
X bi + 1 X
bi + 1
,
+ 2−k .
i+1
i+1
2
2
i=1
i=1
Thus, every x ∈ I has the property 2d~k (x) − ~uk = ~bk , where ~uk = (1, 1, . . . , 1) ∈
~ k (x) = ~bk . We will always use the
{−1, 1}k is a vector with k entries, i.e. X
convention that if k is equal to zero then the corresponding dyadic interval, I, will
be equal to the interval [0, 1). Note that this map is also onto. Now, using ak (·), it
is easy to verify that
~ r (x1 ))Xr+1 (x1 )f~r0 (~x0 ).
(37)
f~r (~x) = ar (X
1
1
Therefore, using the expression (37), we can conclude that
X
~ r (x1 ))Xr+1 (x1 )f~r0 (~x0 )
Hn (~x) =
ar1 (X
1
|~
r |=n
(38)
=
n
X
~ r (x1 ))Xr+1 (x1 )Fr (x2 , . . . , xd ).
ar1 (X
1
1
r1 =0
At this moment, we would like to highlight the importance of the decoupling constraint on the coefficients {αR : R ∈ Dd , |R| = 2−n }. This additional constraint al~ r (·))Xr +1 (·)}n
lows us to isolate the independent random variables {ar1 (X
r1 =0 (this
1
1
fact will be proved in Section 4) in the way it is described in the relation (38). These
~ r (·))Xr +1 (·)}n
two facts, the independence of the random variables {ar1 (X
r1 =0 and
1
1
equation (38), will be crucial for estimating the L∞ norm of Hn (·). This estimate will give us the desired exponent of the positive integer n ≥ 1 in the relation (23) of our main theorem, Theorem 1.9. As we will see in Section 4, with
the aid of Proposition 2.1 and Proposition 2.3, the independent random variables
~ r (·))Xr +1 (·)}n
{ar1 (X
r1 =0 will contribute to the sharp exponent of n by the factor
1
1
so that the full
of 1 and the function Fr1 (·) will contribute by the factor of d−2
2
gain over the trivial bound can be obtained.
4. Proof of Theorem 1.9
First, we prove that the random variables (ak (X~k (·))Xk+1 (·))nk=0 are iid random
variables. We then proceed to the proof of the Theorem 1.9.
Proposition 4.1. The random variables (ak (X~k (·))Xk+1 (·))nk=0 are iid binary random variables on the probability space, ([0, 1), B([0, 1)), P = | · |).
12
DIMITRIOS KARSLIDIS
~ k) =
Proof. First, we show that for fixed 0 ≤ k ≤ n and b0 ∈ {−1, 1}, P (Xk+1 ak (X
1
n
~ k (·))Xk+1 (·))
b0 ) = 2 . We then show that the random variables (ak (X
k=0 are
independent.
Without loss of generality, we can assume that 0 < k ≤ n. Fixing b0 ∈ {−1, 1},
we get
X
~ k ) = b0 ) =
~ k ) = b0 , X
~ k = ~bk )
P (Xk+1 ak (X
P (Xk+1 ak (X
~bk ∈{−1,1}k
X
=
P
Xk+1
~bk ∈{−1,1}k
b0
~ k = ~bk
,X
=
ak (~bk )
!
k
1
=2
2 2
1
= .
2
Therefore, the first assertion has been proven. Now, we prove the second assertion.
~
Using the independence of the random variables (Xi )n+1
i=1 , we get that for any bn+1 ∈
n+1
{−1, 1}
,
k1
~ 0 ) = b1 , . . . , Xn+1 an (X
~ n ) = bn+1 ) = P (X1 = eb1 , . . . , Xn+1 = ebn+1 )
P (X1 a0 (X
n+1
1
=
2
n
Y
~ k ) = bk+1 ),
P (Xk+1 ak (X
=
k=0
where eb1 , . . . , ebn+1 are the deterministic constants depending on the constants
b1 , . . . , bn+1 . The last equality shows the independence of the random variables
~ k (·))Xk+1 (·))n .
(ak (X
k=0
4.1. Proof of Theorem 1.9.
Proof. Using the definition of supremum norm, k · k∞ , and relation (38), we get
n
X
~ r (x1 ) Xr +1 (x1 ) .
(39)
kHn k∞ =
sup
sup Fr1 (~x0 )ar1 X
1
1
~
x0 ∈[0,1]d−1 x1 ∈[0,1] r1 =0
Using, Proposition 2.3 and Proposition 4.1, we will rewrite the inner supremum in
(39). We fix ~x0 = (x2 , x3 , · · · , xd ) ∈ [0, 1]d−1 , and apply both Proposition 4.1 and
~ r (·))Xr +1 (·) and cr = Fr (x2 , x3 , · · · , xd ),
Proposition 2.3 with Ur1 (·) = αr1 (X
1
1
1
1
r1 = 0, 1, . . . , n, in order to get the following equality
(40)
n
n
X
X
0
~ r (x1 ) Xr +1 (x1 ) =
sup Fr1 (~x )ar1 X
|Fr1 (~x0 )|.
1
1
x1 ∈[0,1]
r1 =0
r1 =0
Combining (39) and (40), we get
(41)
kHn k∞ =
sup
n
X
~
x0 ∈[0,1]d−1 r =0
!
|Fr1 (~x0 )|.
ON THE SIGNED SMALL BALL INEQUALITY WITH RESTRICTED COEFFICIENTS
13
Now, it is easy to see from (41) that
(42)
kHn k∞
n
X
|Fr1 |
≥
r! =0
n
X
=
kFr1 kL1 ([0,1]d−1 ) .
r1 =0
L1 ([0,1]d−1 )
Therefore, taking into account inequality (34), the estimate in (42) becomes
kHn k∞ ≥ cd
(43)
n
X
kFr1 kL2 ([0,1]d−1 ) .
r1 =0
Using the orthogonality of Haar functions, we have that
(44)
kFr1 kL2 ([0,1]d−1 ) = k(Sd−1 Fr1 )kL2 ([0,1]d−1 ) & (n − r1 )
d−2
2
.
Finally, putting relations (43) and (44) together, we obtain the desired estimate
in (23) as follows:
kHn k∞ &
n
X
(n − r1 )
d−2
2
d
& n2 .
r1 =0
In summary, the two basic components of the proof are the following:
kHn k∞ =
sup
n
X
~
x0 ∈[0,1]d−1 r =0
!
|Fr1 (~x0 )|
and kFr1 kL1 ([0,1]d−1 ) & kFr1 kL2 ([0,1]d−1 ) ,
where r1 = 0, 1, . . . , n. Results of this type have appeared in related problems in
classical Fourier analysis. An important example is Sidon’s theorem concerning the
behaviour of lacunary Fourier
P series [14, 15]. It states that if f has a Hadamard
lacunary series, e.g., f ∼ ak sin(nk x) with nk+1 /nk > q > 1, then f satisfies
X
kf k∞ &
|ak | and kf k1 ≥ kf k2 .
After the paper was submitted, the author was informed by the referee that similar techniques have appeared in a recent result of Skriganov on discrepancy [16].
Skriganov proves that for any N-point distribution, random digit shifts lead to the
d
conjectured discrepancy inequality kDN k∞ & (log N ) 2 . The referee has pointed out
that the splitting condition of this paper has been replaced there by the randomness
of the digit shifts, but otherwise the methods are rather similar.
References
[1] R. F. Bass, Probability estimates for multiparameter Brownian processes, Ann. Probab. 16
(1988), no. 1, 251-264.
[2] J. Beck, A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution,
Compositio Math., 72 (1989), no. 3, 269-339.
[3] D. Bilyk and M. Lacey, On the small ball inequality in three dimensions, Duke Math. J., 143
(2008), no. 1, 81-115.
[4] D. Bilyk, M. Lacey and A. Vagharshakyan, On the small ball inequality in all dimensions,
J. Funct. Anal., 254 (2008), no. 9, 2470-2505.
[5] D. Bilyk, M. Lacey and A. Vagharshakyan, On the signed small ball inequality, Online Journal
of Analytic Combinatorics, 3 (2008).
14
DIMITRIOS KARSLIDIS
[6] D. Bilyk, M. Lacey, I. Parissis and A. Vagharshakyan, A three-dimensional small ball inequality, “Dependence in Probability, Analysis and Number Theory”, Walter Philip memorial
volume, 73-78, Kendrick Press, Heber City, UT, 2010.
[7] D. Bilyk On Roth’s orthogonal function method in discrepancy theory, Unif. Distrib. Theory
6 (2011), no. 1.
[8] E. Csáki, On small values of the square integral of a multiparameter Wiener process. In:
Statistics and probability (Visegrád, 1982), pp. 1926. Reidel, Dordrecht (1984).
[9] T. Dunker, T. Kühn, M. Lifshits, W. Linde, Metric entropy of the integration operator and
small ball probabilities for the Brownian sheet, C. R. Acad. Sci. Paris Sér. I Math., 326
(1998), no. 3, 347-352.
[10] J. Kuelbs, W. V. Li, Metric entropy and the small ball problem for Gaussian measures, J.
Funct. Anal 116 1, 133-157(1993)
[11] S. Resnick, A Probability Path, Birkhauser,Boston(1998)
[12] K. F. Roth, On irregularities of distribution, Mathimatika, 1 (1954), 73-79.
[13] W. M. Schmidt, Irregularities of distribution VI. Composito Math. 24, 63-74 (1972).
[14] S. Sidon, Verallgemeinerung Emus Satzes über die absolute Konvergenz von Fourierreihen
mit Lücken, Math. Ann. 97. 675-676 (1927)
[15] S. Sidon, Ein Satzüber trigonometrische Polyfonem mit Lg̈ucken und seine Anwendung in
der Theorie der Fourier Reihen. J. Reine Angew, Math. 163, 251-252 (1930)
[16] M. M. Skriganov, Dyadic shift randomization in classical discrepancy theory, available at
http://arxiv.org/abs/1409.1997
[17] M. Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (1994), no.
3, 1331-1354.
[18] C. Thiele,Lectures on Real Analysis,
Bonn 2010/UCLA 2012,
available at
math.ucla.edu/ thiele/lecturenotes/bonnlectures/lecture4.pdf
[19] V. N. Temlyakov, Some inequalities for multivariate Haar polynomials, East Journal on
Approximations, 1 (1995), no. 1, 61-72.
[20] V. N. Temlyakov, An inequality for trigonometric polynomials and its applications for estimating the Kolmogorov widths, East Journal on Approximations, 2 (1996), 253-262.
Dimitrios Karslidis
Department of Mathematics, University of British Columbia
1984 Mathematics Road, Vancouver, BC
Canada V6T 1Z2
E-mail:dimkars@math.ubc.ca