Abstract
The local theory of complex dimensions for real and \(p\)-adic fractal strings describes oscillations that are intrinsic to the geometry, dynamics and spectrum of archimedean and nonarchimedean fractal strings. We aim to develop a global theory of complex dimensions for adèlic fractal strings in order to reveal the oscillatory nature of adèlic fractal strings and to understand the Riemann hypothesis in terms of the vibrations and resonances of fractal strings. We present a simple and natural construction of self-similar \(p\)-adic fractal strings of any rational fractal (i.e., Minkowski) dimension in the closed unit interval \([0,1]\). Moreover, as a first step towards a global theory of complex dimensions for adèlic fractal strings, we construct an adèlic Cantor string in the set of finite adèles \(\mathbb{A}_0\) as an infinite Cartesian product of every \(p\)-adic Cantor string, as well as an adèlic Cantor-Smith string in the ring of adèles \(\mathbb{A}\) as a Cartesian product of the general Cantor string and the adèlic Cantor string.
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Notes
From his paradise Cantor with us unfolded, we hold our breath in awe, knowing we shall not be expelled.
Here, \(|x|_{\infty}\) denotes the usual absolute value of \(x\in \mathbb{Q}^*\).
Two absolute values on \(\mathbb Q\) are said to be equivalent if one is a positive power of the other.
From a technical point of view, both infinite products are really ‘restricted products’.
References
C. Andreae and L. Franke, Georg Cantor: Das erste Diagonalverfahren, Oberstufe Mathematik-Projekt Unendlichkeit, www.rudolf-web.de (2004).
G. F. L. P. Cantor, “Uber unendliche, lineare Punktmannichfaltigkeiten, Part 5,” Math. Ann. 21, 545–591 (1883).
A. Connes, Noncommutative Geometry (Academic Press, San Diego, 1994).
M. du Sautoy, The Music of the Primes (Harper Collins, New York, 2003).
K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, third edition (John Wiley & Sons, Chichester, 2014).
J. F. Fleron, “A note on the history of the Cantor set and Cantor function,” Math. Magaz. 67 (2), 136–140 (1994).
P. Freund and E. Witten, “Adelic string amplitudes,” Phys. Lett. B 199, 191 (1987).
G. W. Gibbons and S. W. Hawking, (eds.), Euclidean Quantum Gravity (World Scientific Publ., Singapore, 1993).
R. Harvey and J. Polking, “Removable singularities of solutions of linear partial differential equations,” Acta Math. 125, 39–56 (1970).
S. W. Hawking and W. Israel, (eds.), General Relativity: An Einstein Centenary Survey (Cambridge Univ. Press, Cambridge, 1979).
H. Herichi and M. L. Lapidus, Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality (World Scientific Publishing, Singapore, 2021).
A. E. Ingham, The Distribution of Prime Numbers (Cambridge Univ. Press, Cambridge, 1932).
M. Kesseböhmer and S. Kombrink, “Fractal curvature measures and Minkowski content for self-conformal subsets of the real line,” Adv. Math. 230, 2474–2512 (2012).
A. Kumar, M. Rani and R. Chugh, “New 5-adic Cantor sets and fractal string,” SpringerPlus 2, 654 (2013).
M. L. Lapidus, “Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture,” Trans. Amer. Math. Soc. 325, 465–529 (1991).
M. L. Lapidus, “Spectral and Fractal Geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function,” in: Differential Equations and Mathematical Physics, Proc. Fourth UAB Internat. Conf., Birmingham, USA, March 1990 (Bennewitz, C., ed.), pp. 151–182 (Academic Press, New York, 1992).
M. L. Lapidus, “Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture,” in: Ordinary and Partial Differential Equations, vol. IV, Proc. 12th Internat. Conf., Dundee, Scotland, UK, June 1992 (Sleeman, B.D., Jarvis, R.J., eds.), Pitman Research Notes in Mathematics Series 289, pp. 126–209 (Longman Sci. and Tech., London, 1993).
M. L. Lapidus, “Analysis on fractals, Laplacian on self-similar sets, noncommutative geometry and spectral dimensions,” Topol. Meth. Nonlin. Anal. 4 (1), 137–195 (1994).
M. L. Lapidus, In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes (Amer. Math. Soc., Providence, R.I., 2008).
M. L. Lapidus, “An overview of complex fractal dimensions: From fractal strings to fractal drums, and back,” in: Horizons of Fractal Geometry and Complex Dimensions (R. G. Niemeyer, E. P. J. Pearse, J. A. Rock, T. Samuel, eds.), Contemp. Math. 731, 143–265 (Amer. Math. Soc., Providence, R.I., 2019).
M. L. Lapidus and H. Lũ’, “Nonarchimedean Cantor set and string,” J. Fixed Point Theory Appl. 3, 181–190 (2008).
M. L. Lapidus and H. Lũ’, “Self-similar \(p\)-adic fractal strings and their complex dimensions,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 1, 167–180 (2009).
M. L. Lapidus and H. Lũ’, “The geometry of \(p\)-adic fractal strings: A comparative survey,” in: Advances in Nonarchimedean Analysis, Proc. 11th Internat. Conf. on \(p\)-Adic Functional Analysis (Clermont-Ferrand, France, July 2010), (J. Araujo, B. Diarra, A. Escassut, eds.), Contemp. Math. 551, 163–206 (Amer. Math. Soc., Providence, R.I., 2011).
M. L. Lapidus, H. Lũ’ and M. van Frankenhuijsen, “Minkowski measurability and exact fractal tube formulas for \(p\)-adic self-similar strings,” in: Fractal Geometry and Dynamical Systems in Pure Mathematics I: Fractals in Pure Mathematics (D. Carf\‘i, M. L. Lapidus, E. P. J. Pearse, M. van Frankenhuijsen, eds.), Contemp. Math. 600, 161–184 (Amer. Math. Soc., Providence, R.I., 2013).
M. L. Lapidus, H. Lũ’ and M. van Frankenhuijsen, “Minkowski dimension and explicit tube formulas for \(p\)-adic fractal strings,” Fractal Fract. 2, 26th paper, 1–30 (2018).
M. L. Lapidus and H. Maier, “The Riemann hypothesis and inverse spectral problems for fractal strings,” J. London Math. Soc. 52 (2), 15–34 (1995).
M. L. Lapidus and E. P. J. Pearse, “A tube formula for the Koch snowflake curve, with applications to complex dimensions,” J. London Math. Soc. 74 (2), 397–414 (2006).
M. L. Lapidus, E. P. J. Pearse and S. Winter, “Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators,” Adv. Math. 227, 1349–1398 (2011).
M. L. Lapidus and C. Pomerance, “The Riemann zeta function and the one-dimensional Weyl-Berry conjecture for fractal drums,” Proc. London Math. Soc. 66 (3) No. 1, 41–69 (1993).
M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Birkhäuser, Boston, 2000).
M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, Springer Monographs in Math. (Springer, New York, 2006).
M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, Springer Monographs in Math. (Springer, New York, 2013).
M. L. Lapidus, A Panorama of Number Theory: From Euler and Riemann to Weil, and Beyond, book in preparation (for Springer/Birkhäuser, 2021).
M. L. Lapidus, G. Radunović and D. Žubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, Springer Monographs in Math. (Springer, New York, 2017).
B. B. Mandelbrot, “Measures of fractal lacunarity: Minkowski content and alternatives,” in: Fractal Geometry and Stochastics, Progress in Probability 37, (C. Bandt, S. Graf, M. Zähle, eds.) (Birkhäuser, Basel, 1995).
L. Nottale, Fractal Spacetime and Microphysics: Towards a Theory of Scale Relativity (World Scientific Publishing, Singapore, 1993).
A. Ostrowski, “Über einige Lösungen der Funktionalgleichung \(\varphi(x)\cdot\varphi(y)=\varphi(xy)\),” Acta Math. (2nd ed.). 41 (1), 271–284 (1916).
G. F. B. Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse,” English translation in Riemann’s Zeta Function (H.M. Edwards, Dover edition, 2001).
H. J. S. Smith, “On the integration of discontinuous functions,” Proc. London Math. Soc. 6 (1), 140–153 (1875).
J. Tate, “Fourier analysis in number fields and Hecke’s zeta functions,” in: Algebraic Number Theory, 305–347 (Cassels, J.W.S., Fröhlich, A., eds.) (Academic Press, New York, 1967).
M. van Frankenhuijsen, The Riemann Hypothesis for Function Fields, London Math. Soc., Student Texts 80 (Cambridge Univ. Press, Cambridge, 2014).
M. van Frankenhuijsen, “The spectral operator and resonances,” in: Horizons of Fractal Geometry and Complex Dimensions (R. G. Niemeyer, E. P. J. Pearse, J. A. Rock, T. Samuel, eds.), Contemp. Math. 731, 115–131 2019.
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics (World Scientific Publishing, Singapore, 1994).
I. V. Volovich, “Number theory as the ultimate physical theory,” CERN-TH.4781/87, July 1987. Published in: \(p\)-Adic Num. Ultrametr. Anal. Appl. 2 (1), 77–87 (2010).
J. A. Wheeler and K. W. Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (Norton, W.W., New York, 1998).
Funding
The work of the first author (MLL) was partially supported by the US National Science Foundation (NSF) under the research grants DMS-0707524 and DMS-1107750, and by the Burton Jones Endowed Chair in Pure Mathematics (of which MLL was the chair holder at the University of California, Riverside, during the completion of this paper).
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Lapidus, M.L., Hùng, L. & van Frankenhuijsen, M. \(p\)-Adic Fractal Strings of Arbitrary Rational Dimensions and Cantor Strings. P-Adic Num Ultrametr Anal Appl 13, 215–230 (2021). https://doi.org/10.1134/S2070046621030043
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DOI: https://doi.org/10.1134/S2070046621030043