Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

REFERENCES

• Berger , M. and Gostiaux , B. 1988 . Differential Geometry. Manifolds, Curves and Surfaces Berlin : Springer–Verlag. . [Berger and Gostiaux 88], English translation
   Google Scholar
• Elkies , N. D. “Rational Points Near Curves and Small Nonzero |x 3 – y2| via Lattice Reduction.” . Proceedings of ANTS–4 . Edited by: Bosma , W. pp. 33 – 63 . Berlin : Springer–Verlag. . [Elkies 00], Lecture Notes in Computer Science 1838
   Google Scholar
• Falconer , K. J. 1990 . Fractal Geometry. Mathematical Foundations and Applications. Chichester , , UK : Wiley. . [Falconer 90]
   Google Scholar
• Feller , W. 1966 . An Introduction to Probability Theory and its Applications Vol. II , New York : Wiley. . [Feller 66]
   Google Scholar
• Frantz , M. December 2001 . “Minkowski Measurability and Lacu– narity of Self–Similar Sets in R,” December , [Frantz 01], Preprint
   Google Scholar
• Frantz , M. March 2004 . “Lacunarity, Minkowski Content, and Self–Similar Sets in R,”. ” . In Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot Edited by: Lapidus , M. L. and van Frankenhuysen , M. March , Providence , RI : Amer. Math. Soc. . [Frantz 04], To appear, Proc. Symposia in Pure Mathematics
   Google Scholar
• Hardy , G. H. and Wright , E. M. 1960 . An Introduction to the Theory of Numbers. Oxford , , UK : Oxford University Press. . [Hardy and Wright 60]
   Google Scholar
• Hastad , J. , Just , B. , Lagarias , J. C. and Schnorr , C. P. 1989 . “Polynomial Time Algorithms for Finding Integer Relations among Real Numbers.” . SIAM J. Com– put. , 18 : 859 – 881 . [Hastad et al. 89]
   Web of Science ®Google Scholar
• Ingham , A. E. 1992 . The Distribution of Prime Numbers. Cambridge , , UK : Cambridge University Press. . [Ingham 92]
   Google Scholar
• Kigami , J. and Lapidus , M. L. 1993 . “Weyl's Problem for the Spectral Distribution of Lapla– cians on p.c.f. Self–Similar Fractals.” . Commun. Math. Phys. , 158 : 93 – 125 . [Kigami and Lapidus 93]
   Web of Science ®Google Scholar
• Lagarias , J. C. 1985 . “The Computational Complexity of Simultaneous Diophantine Approximation Problems.” . SIAM J. Comput. , 14 : 196 – 209 . [Lagarias 85]
   Web of Science ®Google Scholar
• Lalley , S. P. 1989 . “Renewal Theorems in Symbolic Dynamics, with Applications to Geodesic Flows, Noneuclid– ean Tessellations and their Fractal Limits.” . Acta Math. , 163 : 1 – 55 . [Lalley 89]
   Web of Science ®Google Scholar
• Lalley , S. P. 1991 . “Probabilistic Counting Methods in Certain Counting Problems of Ergodic Theory.” . In Er– godic Theory, Symbolic Dynamics and Hyperbolic Spaces Edited by: Bedford , T. , Keane , M. and Series , C. 223 – 258 . Oxford , , UK : Oxford Univ. Press. . [Lalley 91]
   Google Scholar
• Lapidus , M. L. 1993 . “Vibrations of Fractal Drums, the Riemann Hypothesis, Waves in Fractal Media, and the Weyl Berry Conjecture.” . In Ordinary and Partial Differential Equations Edited by: Sleeman , B. D. and Jarvis , R. J. Vol. IV , 126 – 209 . London : Longman Scientific and Technical. . [Lapidus 93], Proc. Twelfth Intern. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series 289
   Google Scholar
• Lapidus , M. L. and Maier , H. 1995 . “The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings.” . J. London Math. Soc. (2) , 52 : 15 – 34 . [Lapidus and Maier 95]
   Web of Science ®Google Scholar
• Lapidus , M. L. and Pomer– ance , C. 1993 . “The Riemann Zeta–Function and the OneDimensional Weyl–Berry Conjecture for Fractal Drums.” . Proc. London Math. Soc. (3) , 66 : 41 – 69 . [Lapidus and Pomerance 93]
   Web of Science ®Google Scholar
• Lapidus , M. L. and van Frankenhuysen , M. 1999 . “Complex Dimensions of Fractal Strings and Oscillatory Phenomena in Fractal Geometry and Arithmetic.” . Contemporary Mathematics , 237 : 87 – 105 . [Lapidus and van Frankenhuysen 99]
   Google Scholar
• Lapidus , M. L. and van Frankenhuysen , M. 2000 . Fractal Geometry and Number Theory (Complex dimensions of fractal strings and zeros of zeta functions) Boston , MA : Birkhauser. . [Lapidus and van Frankenhuysen 00], Research Monographs, (Second revised and enlarged edition to appear in 2004.)
   Google Scholar
• Lapidus , M. L. and van Frankenhuysen , M. 2001 . “A Prime Orbit Theorem for SelfSimilar Flows and Diophantine Approximation.” . Contemporary Mathematics , 290 : 113 – 138 . [Lapidus and van Frankenhuysen 01a]
   Google Scholar
• Lapidus , M. L. and van Frankenhuysen , M. 2001 . “Complex Dimensions of SelfSimilar Fractal Strings and Diophantine Approximation.” Berkeley , CA : Mathematical Sciences Research Institute (MSRI). . [Lapidus and van Frankenhuysen 01b], MSRI Preprint No. 2001–040
   Google Scholar
• Lenstra , A. K. , Lenstra , H. W. Jr. and Lovász , L. 1982 . “Factoring Polynomials with Rational Coefficients.” . Math. Ann. , 261 : 515 – 534 . [Lenstra et al. 82]
   Web of Science ®Google Scholar
• Mandelbrot , B. B. 1983 . The Fractal Geometry of Nature. New York : Freeman. . [Mandelbrot 83]
   Google Scholar
• Ostrowski , A. 1922 . “Bemerkungen zur Theorie der Diophantischen Approximationen.” . Abh. Math. Sem. Hamburg Univ. , 1 : 77 – 98 . [Ostrowski 22]
   Google Scholar
• Parry , W. and Pollicott , M. 1983 . “An Analogue of the Prime Number Theorem and Closed Orbits of Axiom A Flows.” . Annals of Math. , 118 : 573 – 591 . [Parry and Pollicott 83]
   Web of Science ®Google Scholar
• Parry , W. and Pollicott , M. 1990 . “Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics.” . In Astérisque 187 – 188 . Paris : Soc. Math. France. . [Parry and Pollicott 90]
   Google Scholar
• Rössner , C. and Schnorr , C. P. “An Optimal, Stable Continued Fraction Algorithm for Arbitrary Dimension.” . Proceedings of IPCO V (Conference on Integer Programming and Combinatorial Optimization, Vancouver, June 3–5, 1996) . pp. 31 – 43 . Berlin : Springer– Verlag. . [Rössner and Schnorr], Lecture Notes in Computer Science 1084
   Google Scholar
• Schmidt , W. M. 1980 . Diophantine Approximation Berlin : Springer– Verlag. . [Schmidt 80], Lecture Notes in Mathematics, 785
   Google Scholar
• Strichartz , R. S. 1990 . “Fourier Asymptotics of Fractal Measures.” . J. Functional Anal. , 89 : 154187 [Strichartz 90]
   Google Scholar
• Strichartz , R. S. 1990 . “Self–Similar Measures and their Fourier Transforms, I.” . Indiana Univ. Math. J. , 39 : 797 – 817 . [Strichartz 93], II, Trans. Amer. Math. Soc. 336 (1993), 335–361; III, Indiana Univ. Math. J. 42 (1993), 367–411
   Web of Science ®Google Scholar
• van Frankenhuysen , M. 2003 . [van Frankenhuysen 03], Available from World Wide Web http://www.math.rutgers.edu/∼machiel/programs/
   Google Scholar
• Watson , G. N. 1995 . A Treatise on the Theory of Bessel Functions, , Second edition Cambridge , , UK : Cambridge Univ. Press. . [Watson 95], Cambridge Mathematical Library
   Google Scholar
• Whittaker , E. T. and Watson , G. N. 1996 . A Course of Modem Analysis, , Fourth edition Cambridge , , UK : Cambridge Univ. Press. . [Whittaker and Watson 96], Cambridge Mathematical Library
   Google Scholar