1. Riesz Pairs and Feichtinger’s Conjecture
INTERNATIONAL CONFERENCE IN MATHEMATICS AND APPLICATIONS (ICMA - MU 2009)
Wayne Lawton
Department of Mathematics
National University of Singapore

2. Titles Background Equivalences Subject Syndetic Sets Min. Seq.
Symbolic Dynamics
Objectives
Densities
Fat Cantor Sets
Known Results
Power Spectral Measure
New Result
Thue-Morse Min. Seq.
Tower of Hanoi
Thue-Morse Spec. Meas.
Volterra Iteration
MATLAB Code
Thue-Morse Distribution
Thue-Morse Spec. Meas.
Spline Approx. Algorithm
Spline Approx. Distribution
Spline Approx. Spec. Meas.
Distribution Comparison
Binary Tree Model
Binomial Approximation
Hausdorff-Besicovitch Dim.
Thickness of Cantor Sets
Research Questions
References

3. Background Recently there has been considerable interest in two deep
problems that arose from very separate areas of mathematics.
Kadison-Singer Problem (KSP): Does every pure state on the
-subalgebra
admit a unique extension to
arose in the area of operator algebras and has remained
unsolved since 1959 [KS59].
Feichtinger’s Conjecture (FC): Every bounded frame can
be written as a finite union of Riesz sequences.
arose from Feichtinger's work in the area of signal processing
involving time-frequency analysis and has remained unsolved
since it was formally stated in the literature in 2005 [CA05].
[KS59] R. Kadison and I. Singer, Extensions of pure states, Amer. J. Math., 81(1959),
[CA05] P. G. Casazza, O. Christiansen, A. Lindner and R. Vershynin, Frames
and the Feichtinger conjecture, PAMS, (4)133(2005),

4. Equivalences Casazza and Tremain proved ([CA06b], Thm 4.2)
that a yes answer to the KSP is equivalent to FC.
Casazza, Fickus, Tremain, and Weber [CA06a]
explained numerous other equivalences.
[CA06b] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006),
[CA06a] P. G. Casazza, M. Fickus, J. Tremain, and E. Weber, The Kadison-Singer problem in mathematics and engineering, Contemp. Mat., 414, AMS, Providence, RI, 2006, pp

5. Subject Feichtinger’s Conjecture for Exponentials (FCE):
of this talk is the following special case of FC:
Feichtinger’s Conjecture for Exponentials (FCE):
For every non-trivial measurable set
the sequence
is a finite union of Riesz sequences*.
*If
is a Riesz sequence if there exists
such that every trigonometric polynomial
(with frequencies in )
satisfies

6. Syndetic Sets and Minimal Sequences
is syndetic if there exists a positive integer
with
is a minimal sequence if its orbit closure
is a minimal closed shift-invariant set.
These are core concepts in symbolic topological dynamics [GH55]
[GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.

7. Symbolic Dynamics Connection
Theorem 1.1 [LA09] For measurable
the
following conditions are equivalent:
is a finite union of Riesz sequences. 1.
such that
There exists a syndetic set 2.
is a Riesz sequence.
such that
There exists a nonempty set 3.
is a minimal sequence and
is a Riesz sequence.
[LA09] Minimal Sequences and the Kadison-Singer Problem, accepted BMMSS

8. Objectives Near Term: Characterize Riesz pairs (pairs such that
is a Riesz basis)
Long Term: Contribute to the understanding of FCE
and hopefully to FC and the KS problem.

9. Densities Lower and Upper Beurling Lower and Upper Asymptotic
and Separation

10. Fat Cantor Sets
Smith–Volterra–Cantor set (SVC) or the fat Cantor set is an
example of a set of points on the real line R that is nowhere
dense (in particular it contains no intervals), yet has positive
measure. The Smith–Volterra–Cantor set is named after the
mathematicians Henry Smith, Vito Volterra and Georg Cantor.
The Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].
The process begins by removing the middle 1/4 from the interval [0, 1] to obtain                   
The following steps consist of removing subintervals of width 1/22n from the middle of each of the 2n−1 remaining intervals. Then remove the intervals (5/32, 7/32) and (25/32, 27/32) to get                                 

11. Known Results [LA09] Corollary 1.1 [MV74] Corollary 2
[CA01] Theorem 2.2
(never the case if S is a Cantor set)
[BT87] Res. Inv. Thm.
[BT91] Theorem 4.1
(occurs if S is a boring fat Cantor set)
[LA09] Theorem 2.1
[MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974),
[CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001),
[BT87] J. Bourgain and L. Tzafriri, Invertibility of "large" submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Mathematics, (2) 57 (1987),
[BT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43.

12. Power Spectral Measure
is wide sense stationary if
Definition A function
and
exist.
Since
is positive definite the Bochner-Herglotz Theorem
implies there exists a positive measure on
such that
Theorem (Khinchin, Wiener, Kolmogorov)

13. New Result is Theorem If is a fat Cantor set and if such that
is wide sense stationary and
and for all
there exists a closed set
such that
and
then
is not a RP.
Proof
Define
then
and

14. Thue-Morse Minimal Sequence
can be constructed for nonnegative
1. through substitutions 001,110
2. through concatenations 00|1 0|1|10  0|1|10|1001  3.
4. solution of Tower of Hanoi puzzle
The Thue–Morse sequence was first studied by Eugene Prouhet in 1851, who applied
it to number theory. However, Prouhet did not mention the sequence explicitly; this was
left to Axel Thue in 1906, who used it to found the study of combinatorics on words.
The sequence was only brought to worldwide attention with the work of Marston Morse
in 1921, when he applied it to differential geometry. The sequence has been discovered
independently many times, not always by professional research mathematicians; for
example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in
1929 in an application to chess: by using its cube-free property (see above), he showed
how to circumvent a rule aimed at preventing infinitely protracted games by declaring
repetition of moves a draw.

16. Thue-Morse Spectral Measure
can be represented using a Riesz product
[KA72] Theorem 2nd term is purely singular continuous and
has dense support.
Corollary Let
For every
there exists a fat Cantor set
such that
and
is not a RP.
[KA72] S. Kakutani, Strictly ergodic symbolic dynamical systems. In Proc. 6th Berkeley Symp. On Math. Stat. and Prob., eds. Le Cam L. M., Neyman J. and Scott E. El., UC Press, 1972, pp

17. Volterra Iteration that approximates the cumulative distribution
is given by
and is a weak contraction with respect to the total variation norm [BA08] and hence it converges uniformly to
M. Baake and U. Grimm, The singular continuous diffraction measure of the Thue-Morse chain, J. Phys. A: Math. Theor. 41 (2008) (6pp) , arXiv: v2

18. MATLAB CODE function [x,F] = Volterra(log2n,iter)%
n = 2^log2n;
dx = 1/n;
x = 0:dx:1-dx;
S = sin(pi*x/2).^2;
F = x;
for k = 1:iter
dF = F - [0 F(1:n-1)];
P = S.*dF;
I = cumsum(P);
F(1:n/2) = I(1:2:n);
F(n/2+1:n) = 1 - F(n/2:-1:1);
end

19. Thue-Morse Distribution 20 iterations

20. Thue-Morse Spectral Measure

21. Spline Approximation Algorithm
Is obtained by replacing
is given by
also converges uniformly to an approximation to

22. Spline Approx. Distribution (20 iterations)

23. Spline Approx. Spectral Measure

24. Distribution Comparison

25. Binary Tree Model

26. Binomial Approximation
For every
and
the intervals that
contribute
hence
are those with m a’s and (n-m) b’s with
so the fraction of these dyadic intervals is

27. Hausdorff-Besicovitch Dimension
dimensional H. content of a subset
Theorem For the approximate support of
therefore
S. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions". Mathematische Annalen 101 (1929).
S. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions". J. London Mathematical Society 12 (1937).
F. Hausdorff (March 1919). "Dimension und äußeres Maß". Mathematische Annalen 79 (1–2): 157–179.

28. Thickness of Cantor Sets
Ordered Derivation
Thickness
[AS99] Thm 2.4 Let
be Cantor sets. Then
contains an interval.
[AS99] S. Astels, Cantor sets and numbers with restricted partial quotients, TAMS, (1)352(1999),

29. Research Questions Clearly fat Cantor sets have Hausdorff dim =1
and thickness = 1. What are these parameters for
approximate supports of spectral measures of the
Thue-Morse and related sequences?
2. How are these properties related to multifractal
properties of the TM spectral measure [BA06]?
3. How are these parameters related to the Riesz
properties of pairs
4. What happens for gen. Morse seq. [KE68]?
[BA06] Zai-Qiao Bai, Multifractal analysis of the spectral measure of the
Thue-Morse sequence: a periodic orbit approach, J. Phys. A: Math. Gen.
39(2006)
[KE68] M. Keane, Generalized Morse sequences, Z.
Wahrscheinlichkeitstheorie verw. Geb. 10(1968),

30. References
J. Anderson, Extreme points in sets of positive linear maps on B(H), J. Func. Anal. 31(1979),
M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940).
H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925), ;47(1926),
O. Christenson, An Introduction to Frames and Riesz Bases, Birkhauser, 2003.
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981.
H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier property in discrete crossed products, J. Funct. Anal. 69 (1986),
H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986),
N. Weaver, The Kadison-Singer problem in discrepancy theory, preprint