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 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

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. 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]. Kadison-Singer Problem (KSP): Does every pure state on the -subalgebraadmit 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. [KS59] R. Kadison and I. Singer, Extensions of pure states, Amer. J. Math., 81(1959), 547-564. [CA05] P. G. Casazza, O. Christiansen, A. Lindner and R. Vershynin, Frames and the Feichtinger conjecture, PAMS, (4)133(2005), 1025-1033.

4. Equivalences Casazza and Tremain proved ([CA06b], Thm 4.2) that a yes answer to the KSP is equivalent to FC. [CA06b] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006), 2032-2039. Casazza, Fickus, Tremain, and Weber [CA06a] explained numerous other equivalences. [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. 299-355.

5. Subject Feichtinger’s Conjecture for Exponentials (FCE): of this talk is the following special case of FC: 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 integerwith is a minimal sequence if its orbit closure These are core concepts in symbolic topological dynamics [GH55] is a minimal closed shift-invariant set. [GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.

7. Symbolic Dynamics Connection the 1. following conditions are equivalent: Theorem 1.1 [LA09] For measurable is a finite union of Riesz sequences. 2. There exists a syndetic set is a Riesz sequence. such that 3. There exists a nonempty set such that is a minimal sequence and is a Riesz sequence. [LA09] Minimal Sequences and the Kadison-Singer Problem, accepted BMMSS http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

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

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

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 nowherereal linenowhere densedense (in particular it contains no intervals), yet has positiveintervals measuremeasure. The Smith–Volterra–Cantor set is named after the mathematiciansmathematicians Henry Smith, Vito Volterra and Georg Cantor.Henry SmithVito VolterraGeorg Cantor http://en.wikipedia.org/wiki/File:Smith-Volterra-Cantor_set.svg http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf The Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].unit interval 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 1/2 2n from the middle of each of the 2 n−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 [MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82. [CA01] Theorem 2.2 (never the case if S is a Cantor set) [CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54. [BT87] Res. Inv. Thm. [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),137-224. [BT91] Theorem 4.1 (occurs if S is a boring fat Cantor set) [BT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43. [LA09] Theorem 2.1

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

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

14. Thue-Morse Minimal Sequence 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 wasnumber theory left to Axel Thue in 1906, who used it to found the study of combinatorics on words.Axel Thuecombinatorics on words The sequence was only brought to worldwide attention with the work of Marston MorseMarston Morse in 1921, when he applied it to differential geometry. The sequence has been discovereddifferential geometry independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it inMax Euwechess grandmasterteacher 1929 in an application to chess: by using its cube-free property (see above), he showedchess how to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw. http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_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 http://www.jstor.org/pss/2974693

16. Thue-Morse Spectral Measure [KA72] S. Kakutani, Strictly ergodic symbolic dynamical systems. In Proc. 6 th Berkeley Symp. On Math. Stat. and Prob., eds. Le Cam L. M., Neyman J. and Scott E. El., UC Press, 1972, pp. 319-326. can be represented using a Riesz product [KA72] Theorem 2 nd term is purely singular continuous and has dense support. Corollary Let For everythere exists a fat Cantor setsuch that andis not a RP.

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) 422001 (6pp), arXiv:0809.0580v2arXiv:0809.0580v2

18. MATLAB CODE function [x,F] = Volterra(log2n,iter) % 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 approximationto

22. Spline Approx. Distribution (20 iterations)

23. Spline Approx. Spectral Measure

24. Distribution Comparison

25. Binary Tree Model

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

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

28. Thickness of Cantor Sets [AS99] S. Astels, Cantor sets and numbers with restricted partial quotients, TAMS, (1)352(1999), 133-170. Thickness Ordered Derivation [AS99] Thm 2.4 Let contains an interval. be Cantor sets. Then

29. Research Questions 1.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? 3. How are these parameters related to the Riesz properties of pairs [KE68] M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie verw. Geb. 10(1968),335-353 4. What happens for gen. Morse seq. [KE68]? 2. How are these properties related to multifractal properties of the TM spectral measure [BA06]? [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) 10959-10973.

30. References J. Anderson, Extreme points in sets of positive linear maps on B(H), J. Func. Anal. 31(1979), 195-217. H. Bohr, Zur Theorie der fastperiodischen Funktionen I,II,III. Acta Math. 45(1925),29-127;46(1925),101-214;47(1926),237-281 M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940). 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), 121-140. H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986), 355-374. N. Weaver, The Kadison-Singer problem in discrepancy theory, preprint