The document discusses lower bounds for the star discrepancy of point sets and sequences, presenting both known results and conjectures on the corresponding discrepancies based on collaborations with Gerhard Larcher and Roswitha Hofer. It outlines the historical context of discrepancy theory, links it to quasi-Monte Carlo integration, and provides theorems demonstrating the growth of star discrepancy with respect to various dimensions. The document also raises open problems regarding the extension of findings to higher dimensions and the connection with lacunary trigonometric products.

1. LOWER BOUNDS FOR THE
DISCREPANCY OF POINT
SETS AND SEQUENCES
Based on joint works with Gerhard Larcher and Roswitha Hofer.
Florian Puchhammer
August 31, 2017
Université de Montréal, Canada

2. A BRIEF GUIDE TO
DISCREPANCY THEORY

3. Star discrepancy – Definition
How well can we approximate uniform distribution by an N-point set
P ⊆ [0, 1)d
(or sequence S)?
Definition (Discrepancy function, Star discrepancy)
For x = (x1, x2, . . . , xd) ∈ [0, 1)d
we define the discrepancy function
of P as well as its star discrepancy as
DN (x, P) = # (P ∩ [0, x)) − Nx1x2 · · · xd,
D∗
N (P) = DN (·, P) ∞ .
Analogously, for sequences S = (sn)n≥0 we consider
DN (x, S) = # ({sn : 0 ≤ n < N} ∩ [0, x)) − Nx1x2 · · · xd,
D∗
N (S) = DN (·, S) ∞ .
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4. Important connections I
Quasi-Monte Carlo (QMC) integration
Quadrature rules with nodes P = {p1, . . . , pN } ⊆ [0, 1)d
Id(f) =
[0,1)d
f(x) dx ≈
1
N
N
k=1
f(pk) =: QN,P (f).
Koksma–Hlawka inequality: For f : [0, 1]d
→ R with bounded
variation we have,
|Id(f) − QN,P (f)| ≤ cf
D∗
N (P)
N
, cf > 0.
D∗
N (P) determines the quality of this approximation.
Specific upper bounds
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5. Important connections II
Uniform distribution modulo one
Approximation of uniform distribution by deterministic
sequences.
The sequence S is u.d. mod 1 iff
lim
N→∞
D∗
N (S)
N
= 0.
How uniformly can a sequence be distributed?
General lower bounds
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6. LOWER BOUNDS FOR THE
STAR DISCREPANCY

7. Outline
Lower bounds for the star discrepancy
(Comprehensive) historic outline.
Sequences in the unit interval.
Point sets in the unit cube.
General principle
If
D∗
N (P) A(N) for all P ⊆ [0, 1)d+1
then
D∗
N (S) A(N) for all S ⊆ [0, 1)d
.
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8. Historic outline I
Irregularities of distribution: The star discrepancy grows in N.
Theorem (Roth 1954)
For all d ≥ 2, every N point set satisfies
D∗
N (P) ≥ DN (P, ·) 2 d (log N)(d−1)/2
.
Only sharp for L2-discrepancy.
Incorporates tools from harmonic analysis (Haar function
system).
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9. Historic outline II
Theorem (Schmidt 1972, Halász 1981)
Every sequence S in the unit interval satisfies
D∗
N (S) log N.
This estimate is sharp (cf. van der Corput 1935).
G. Halász proved this bound for point sets P ⊆ [0, 1)2.
Refinement of Roth’s (harmonic) approach.
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10. Historic outline III
Halász’ approach not extendible to d ≥ 3 due to shortfall of
certain orthogonality properties.
Theorem (Beck 1989)
For all N-point sets P ⊆ [0, 1)3 and all ε > 0 we have
D∗
N (P) ε log N · (log log N)1/8−ε
.
First significant improvement for d = 3 in 35 years.
Builds upon Halász’ ideas.
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11. Historic outline IV
Theorem (Bilyk, Lacey, Vagharshakayan 2008)
For all d ≥ 3 there exists ηd > 0 such that for every P ⊆ [0, 1)d,
#P = N, we have
D∗
N (P) d (log N)(d−1)/2+ηd
.
As a matter of fact, they mainly focused on the small ball
inequality.
Used new analytic tools (Littlewood–Paley inequalities,
exponential Orlicz spaces) to improve upon Beck.
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12. Historic outline V
For specific point sets the best upper bounds are still of
magnitude (log N)d−1.
Conjecture
Every d-dimensional N-point set P or sequence S in [0, 1)d is
subject to
D∗
N (P) d (log N)d−1
, D∗
N (S) d (log N)d
, or
D∗
N (P) d (log N)d/2
, D∗
N (S) d (log N)(d+1)/2
, or
D∗
N (P) d (log N)(d−1)/2+(d−1)/d
, D∗
N (S) d (log N)d/2+d/(d+1)
.
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13. LOWER BOUNDS FOR THE
STAR DISCREPANCY
Sequences in the unit interval

14. Star discrepancy constant
In the case d = 1 recall
D∗
N (S) log N.
Growth rate is known.
What is the optimal implicit constant?
Definition (Star discrepancy constant)
The one-dimensional star discrepancy constant is defined as
c∗
= inf
S∈[0,1)N
lim sup
N→∞
D∗
N (S)
log N
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15. Main result for d = 1
Theorem (Larcher, P. 2016; Ostromoukhov 2009)
The star discrepancy constant is bounded by
0.0656646 . . . ≤ c∗
≤ 0.222 . . .
Strategy for the lower bound
Link c∗ to the L1-norm of a pertinent auxiliary function
f ∈ F.
Find minimizer of the L1-norm over F independent of N.
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16. The uniform lower bound
Fix S = (sn)n≥1 and let N = at, t ∈ N.
By an inductive argument we obtain the lemma below.
Lemma
Let
f(x) = max
n∈{at−at−1+1,...,at}
Dn(S, x) − max
n∈{1,...,at−1}
Dn(S, x).
Under the assumption that f 1 ≥ b(a) we have
c∗
≥
b(a)
2 log a
.
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17. The minimization problem
f(x) = max
n∈{at−at−1+1,...,at}
Dn(S, x) − max
n∈{1,...,at−1}
Dn(S, x)
The function f lies in a certain function space F.
This space F is large enough so that we can define f∗ ∈ F
via
f∗
1 = min
g∈F
g 1.
New goal: Find b(a) such that f∗
1 ≥ b(a).
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18. Estimation of the solution
Lemma
The function f∗ comprises exactly at−1 parts Q0, at−1(a − 2)
parts Q1, and parts Q
(n)
2 , 1 ≤ n < at−1.
Each such part is characterized by an interval [α, β] with
f∗(α) = f∗(β) = 0, one discontinuity at some point γ, and the
behavior of the slope of f∗ on [α, β].
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19. Finalizing the result
After further minimization steps (e.g., length of the parts) we
obtain that for all a ∈ [3, 3.7) we have
f∗
1 ≥ b(a),
b(a) =
(a − 2) 12a + 9 + (a − 2)(4a − 3) log 1 + 1
a−2
a a − 1
2
2
3 + (a − 2) log 1 + 1
a−2
and, finally,
c∗
≥
b(a)
2 log a
= 0.0656646 . . .
for a specific choice of a.
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20. Open questions
Within the proof we determine the shape of f∗ very clearly.
We have a loose link between indices of S = (sn)n and
discontinuities of f∗.
In addition, only the ordering of the different parts is
undetermined (not entirely).
Open problem
Given the above information, can we derive a new construc-
tion principle for sequences in [0, 1) with a particularly small
discrepancy?
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21. LOWER BOUNDS FOR THE
STAR DISCREPANCY
Point sets in the unit cube

22. Overview
Theorem (Bilyk, Lacey 2008; P. 2016)
For all N-point sets P ⊆ [0, 1)3 and all δ > 0 the star discrep-
ancy is bounded by
D∗
N (P) (log N)1+η
, with η =
1
32 + 4
√
41
− δ.
To illustrate the proof we adhere to the following guidline:
Highlight Halász’ idea in dimension d = 2.
Explain shortfalls in higher dimensions and the strategies
of Beck and Bilyk & Lacey.
Touch upon the core issue: Study of coincidences.
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23. Halász’ approach I
Main idea
Use auxiliary function Φ.
Φ should have a small L1-norm, i.e. Φ 1 1.
At the same time we require Φ, DN (P, ·) log N.
Then:
D∗
N (P) = DN (P, ·) ∞ Φ, DN (P, ·) log N.
This type of behavior can be achieved by sums of products of
signed Haar functions.
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24. Halász’ approach II
The Haar function system
Definition (Dyadic interval, Haar function)
The set of dyadic intervals is given by
D = [a2−k
, (a + 1)2−k
) : k ∈ N and 0 ≤ a < 2k
.
Subdivide each J ∈ D into J = Jl ∪ Jr and define hJ = −1Jl
+ 1Jr
.
For R = J1 × J2 × · · · × Jd ∈ Dd
and x = (x1, x2, . . . , xd) ∈ [0, 1)d
we set
hR(x) = hJ1
(x1)hJ2
(x2) · · · hJd
(xd).
Notice that
EhR = 0 and h2
R = 1R.
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25. Halász’ approach III
Main advantage of Haar functions
Proposition (Product rule)
Let R1 = R2 ∈ Dd
with R1 ∩ R2 = ∅. Under the assumption that the
sides of R1 and R2 are distinct we have
hR1 hR2 = ±hR1∩R2 .
The product rule indicates orthogonality.
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26. Halász’ approach IV
Definition
Hyperbolic vectors: Hd
n = {r ∈ Nd
: r 1 = n}.
Two or more hyperbolic vectors have a coincidence if they agree
in one coordinate. They are strongly distinct otherwise.
An r-function with parameter r = (r1, r2, . . . , rd) ∈ Hd
n is defined
as
fr =
R=(R1,...,Rd)∈Dd
|Rt|=2−rt
α(R)hR, α(R) ∈ {−1, 1}.
The product rule applies within a product frfs, if the vectors
r, s ∈ Hd
n are strongly distinct.
Every r ∈ H2
n is of the form r = (k, n − k).
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27. Halász’ approach V
The auxiliary function
Choose n ≈ log2 N. The function Φ is defined as the Riesz
product
Φ =
n
k=0
1 + γf(k,n−k) −1 = γ
n
k=0
f(k,n−k) +Φ>n, 0 < γ < 1.
Key observation: In a product
f(k1,n−k1)f(k2,n−k2) · · · f(kl,n−kl), k1 < k2 < . . . < kl,
the coordinates of the dyadic rectangles cannot coincide.
We have Φ 1 ≤ 2 and by special choice of α(R)’s
DN (P, ·), Φ n.
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28. Extension to higher dimensions
. . . is not possible on the fly.
Problem: The key observation ceases to hold in d = 3
already.
Solution: Modify the auxiliary function and deal with
coincidences.
Beck: Use graph theory for coincidences and shorten
Riesz product to (log log N)ε factors.
B. & L.: Like Beck, but take (log N)ε factors. Therefore,
they needed new tools:
Littlewood–Paley inequalities,
exponential Orlicz spaces,
conditional expectation arguments.
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29. The study of coincidences
We need to study long collections of coincidal hyperbolic
vectors.
ε acts as a controlling parameter (η = ε/4).
Coincidences are described by two-colored graphs
G = (V, E2, E3).
< < < < <
• = • = • ◦ ◦ ◦
◦ ◦ • = • = • = •
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30. Open problems I
Finally, we have
Φ 1 1, Φ, DN (P, ·) log N for all ε <
1
8 +
√
41
.
Open problem
Can this result be extended to d ≥ 4?
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31. Open problems II
The (S)SBI shows formal (d = 2) and the heuristic (d ≥ 3)
connections to discrepancy and well-distributed point sets.
Proofs for the (S)SBI work analogously, but with a different
function on the dual side of Φ in the inner product.
In case of the SSBI, the arguments are significantly
simpler (avoid study of long coincidences and work in all
dimensions).
Open problem
Are the arguments for the SSBI somehow transferable to dis-
crepancy estimates?
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32. OFF-TOPIC BUT ONLY A LITTLE...

33. Lacunary trigonometric products
R. Hofer and I investigated perturbed Halton–Kronecker
sequence
Discrepancy bounds are in one-to-one relation with
lacunary trigonometric products
Πr,γ(α) =
r−1
j=0
cos 2j
απ +
γj
2
π , γ ∈ {0, 1}N
.
Found sharp upper bounds in the case
γ = (10 . . . 0 10 . . . 0 . . .)
for arbitrary α and tight metric bounds (generalizations of
Gel’fond 1967/1968 and Fouvry & Mauduit 1996).
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34. Open problems
Would be interesting to extend results to other periodic
γ ∈ {0, 1}N
, e.g.
r/3−1
j=0
sin 2j
απ cos 2j+1
απ sin 2j+2
απ
Experiments indicate that bounds only depend on the density of
1s.
No reason to doubt that strategy for obtaining estimates is
essentially the same.
Problem
Find sharp general as well as tight metric bounds for Πr,γ(α) for
periodic γ.
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