Fourier transforms of Gibbs measures for the Gauss map

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• 作者：Thomas Jordan ; Tuomas Sahlsten
• 刊名：Mathematische Annalen
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：364
• 期：3-4
• 页码：983-1023
• 全文大小：680 KB
• 参考文献：1.Alkauskas, G.: The Minkowski \(?(x)\) function and Salem’s problem. C. R. Math. Acad. Sci. Paris 350(3–4), 137–140 (2012)MathSciNet CrossRef MATH
  2.Baker, R.: Metric number theory and the large sieve. J. Lond. Math. Soc. (2) 24(1), 34–40 (1981)
  3.Bluhm, C.: On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat. 36(2), 307–316 (1998)MathSciNet CrossRef MATH
  4.Bluhm, C.: Liouville numbers, Rajchman measures, and small Cantor sets. Proc. Am. Math. Soc. 128(9), 2637–2640 (2000)MathSciNet CrossRef MATH
  5.Canto-Martín, F., Hedenmalm, H., Montes-Rodríguez, A.: Perron–Frobenius operators and the Klein–Gordon equation. J. Eur. Math. Soc. (JEMS) 16(1), 31–66 (2014)MathSciNet CrossRef MATH
  6.Davenport, H., Erdős, P., LeVeque, W.J.: On Weyl’s criterion for uniform distribution. Michigan Math. J. 10, 311–314 (1963)MathSciNet CrossRef MATH
  7.Durrett, R.: Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 4th edn. Cambridge University Press, Cambridge (2010)CrossRef
  8.Falconer, K.J.: Techniques in Fractal Geometry. Wiley, Chichester (1997)MATH
  9.Good, I.: The fractional dimensional theory of continued fractions. Math. Proc. Camb. Philos. Soc. 37(3), 199–228 (1941)MathSciNet CrossRef MATH
  10.Hochman, M., Shmerkin, P.: Equidistribution from fractals. Invent. Math. (2013) (preprint at arXiv:​1302.​5792 )
  11.Hofbauer, F.: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval. Stud. Math. 103(2), 191–206 (1992)MathSciNet MATH
  12.Jarník, V.: Zur metrischen theorie der diophantischen approximationen. Prace Mat.-Fiz. 36(1), 91–106 (1928–1929)
  13.Kaufman, R.: Continued fractions and Fourier transforms. Mathematika 27(2), 262–267 (1980)MathSciNet CrossRef MATH
  14.Kaufman, R.: On the theorem of Jarník and Besicovitch. Acta Arith. 39(3), 265–267 (1981)MathSciNet MATH
  15.Kaufman, R.: \(M\) -sets and measures. Ann. Math. (2) 135(1), 125–130 (1992)
  16.Kesseböhmer, M., Stratmann, B.O.: Fractal analysis for sets of non-differentiability of Minkowski’s question mark function. J. Number Theory 128(9), 2663–2686 (2008)MathSciNet CrossRef MATH
  17.Khinchin, A.: Continued Fractions. Dover Publications Inc., Mineola (1997, Russian edition) (with a preface by B. V. Gnedenko, reprint of the 1964 translation)
  18.Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321(2), 505–524 (1990)MathSciNet CrossRef MATH
  19.Kifer, Y., Peres, Y., Weiss, B.: A dimension gap for continued fractions with independent digits. Israel J. Math. 124, 61–76 (2001)MathSciNet CrossRef MATH
  20.Kinney, J.R.: Note on a singular function of Minkowski. Proc. Am. Math. Soc. 11, 788–794 (1960)MathSciNet CrossRef MATH
  21.Lopes, A.: Entropy and large deviation. Nonlinearity 3(2), 527–546 (1990)MathSciNet CrossRef MATH
  22.Lyons, R.: Seventy years of Rajchman measures. In: Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), J. Fourier Anal. Appl., pp. 363–377 (1995)
  23.Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge (1995)CrossRef MATH
  24.Mattila, P.: Hausdorff dimension, projections, and the Fourier transform. Publ. Mat. 48(1), 3–48 (2004)MathSciNet CrossRef MATH
  25.Mauldin, R.D., Urbański, M.: Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, vol. 148. Cambridge University Press, Cambridge (2003)
  26.Minkowski, H.: Geometrie der zahlen. Gesammelte Abhandlungen, vol. 2 (1911)
  27.Montgomery, H.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, vol. 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1994)
  28.Persson, T.: On a problem by R. Salem concerning minkowski’s question mark function (2015, preprint). arXiv:​1501.​00876
  29.Pollington, A., Velani, S.: On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture. Acta Math. 185(2), 287–306 (2000)MathSciNet CrossRef MATH
  30.Queffélec, M., Ramaré, O.: Analyse de Fourier des fractions continues à quotients restreints. Enseign. Math. (2) 49(3–4), 335–356 (2003)
  31.Salem, R.: On some singular monotonic functions which are strictly increasing. Trans. Am. Math. Soc. 53, 427–439 (1943)MathSciNet CrossRef MATH
  32.Sarig, O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217(3), 555–577 (2001)MathSciNet CrossRef MATH
  33.Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Am. Math. Soc. 131(6), 1751–1758 (2003, electronic)
  34.Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Trans. Am. Math. Soc. 236, 121–153 (1978)MathSciNet CrossRef MATH
  35.Wolff, T.H.: Lectures on Harmonic Analysis, University Lecture Series, vol. 29. American Mathematical Society, Providence (2003) (with a foreword by Charles Fefferman and preface by Izabella Łaba, Edited by Łaba and Carol Shubin)
  36.Yakubovich, S.: The affirmitive solution to Salem’s problem revisited (2015, preprint). arXiv:​1501.​00141
  37.Young, L.-S.: Large deviations in dynamical systems. Trans. Am. Math. Soc. 318(2), 525–543 (1990)MathSciNet MATH
  38.Yuri, M.: Large deviations for countable to one Markov systems. Commun. Math. Phys. 258(2), 455–474 (2005)MathSciNet CrossRef MATH
  
• 作者单位：Thomas Jordan (1)
  Tuomas Sahlsten (2)
  
  1. School of Mathematics, University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, England, UK
  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807

文摘

We investigate under which conditions a given invariant measure \(\mu \) for the dynamical system defined by the Gauss map \(x \mapsto 1/x \,\,{\mathrm {mod}}\,1\) is a Rajchman measure with polynomially decaying Fourier transform $$\begin{aligned} |\widehat{\mu }(\xi )| = O(|\xi |^{-\eta }), \quad \text {as} \quad |\xi | \rightarrow \infty . \end{aligned}$$We show that this property holds for any Gibbs measure \(\mu \) of Hausdorff dimension greater than 1 / 2 with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than 1 / 2 on badly approximable numbers, which extends the constructions of Kaufman and Queffélec–Ramaré. Our main result implies that the Fourier–Stieltjes coefficients of the Minkowski’s question mark function decay to 0 polynomially answering a question of Salem from 1943. As an application of the Davenport–Erdős–LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman–Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents. Mathematics Subject Classification 42A38 11K50 37C30 60F10 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (38) References1.Alkauskas, G.: The Minkowski \(?(x)\) function and Salem’s problem. C. R. Math. Acad. Sci. Paris 350(3–4), 137–140 (2012)MathSciNetCrossRefMATH2.Baker, R.: Metric number theory and the large sieve. J. Lond. Math. Soc. (2) 24(1), 34–40 (1981)3.Bluhm, C.: On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat. 36(2), 307–316 (1998)MathSciNetCrossRefMATH4.Bluhm, C.: Liouville numbers, Rajchman measures, and small Cantor sets. Proc. Am. Math. Soc. 128(9), 2637–2640 (2000)MathSciNetCrossRefMATH5.Canto-Martín, F., Hedenmalm, H., Montes-Rodríguez, A.: Perron–Frobenius operators and the Klein–Gordon equation. J. Eur. Math. Soc. (JEMS) 16(1), 31–66 (2014)MathSciNetCrossRefMATH6.Davenport, H., Erdős, P., LeVeque, W.J.: On Weyl’s criterion for uniform distribution. Michigan Math. J. 10, 311–314 (1963)MathSciNetCrossRefMATH7.Durrett, R.: Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 4th edn. Cambridge University Press, Cambridge (2010)CrossRef8.Falconer, K.J.: Techniques in Fractal Geometry. Wiley, Chichester (1997)MATH9.Good, I.: The fractional dimensional theory of continued fractions. Math. Proc. Camb. Philos. Soc. 37(3), 199–228 (1941)MathSciNetCrossRefMATH10.Hochman, M., Shmerkin, P.: Equidistribution from fractals. Invent. Math. (2013) (preprint at arXiv:​1302.​5792)11.Hofbauer, F.: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval. Stud. Math. 103(2), 191–206 (1992)MathSciNetMATH12.Jarník, V.: Zur metrischen theorie der diophantischen approximationen. Prace Mat.-Fiz. 36(1), 91–106 (1928–1929)13.Kaufman, R.: Continued fractions and Fourier transforms. Mathematika 27(2), 262–267 (1980)MathSciNetCrossRefMATH14.Kaufman, R.: On the theorem of Jarník and Besicovitch. Acta Arith. 39(3), 265–267 (1981)MathSciNetMATH15.Kaufman, R.: \(M\)-sets and measures. Ann. Math. (2) 135(1), 125–130 (1992)16.Kesseböhmer, M., Stratmann, B.O.: Fractal analysis for sets of non-differentiability of Minkowski’s question mark function. J. Number Theory 128(9), 2663–2686 (2008)MathSciNetCrossRefMATH17.Khinchin, A.: Continued Fractions. Dover Publications Inc., Mineola (1997, Russian edition) (with a preface by B. V. Gnedenko, reprint of the 1964 translation)18.Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321(2), 505–524 (1990)MathSciNetCrossRefMATH19.Kifer, Y., Peres, Y., Weiss, B.: A dimension gap for continued fractions with independent digits. Israel J. Math. 124, 61–76 (2001)MathSciNetCrossRefMATH20.Kinney, J.R.: Note on a singular function of Minkowski. Proc. Am. Math. Soc. 11, 788–794 (1960)MathSciNetCrossRefMATH21.Lopes, A.: Entropy and large deviation. Nonlinearity 3(2), 527–546 (1990)MathSciNetCrossRefMATH22.Lyons, R.: Seventy years of Rajchman measures. In: Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), J. Fourier Anal. Appl., pp. 363–377 (1995)23.Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge (1995)CrossRefMATH24.Mattila, P.: Hausdorff dimension, projections, and the Fourier transform. Publ. Mat. 48(1), 3–48 (2004)MathSciNetCrossRefMATH25.Mauldin, R.D., Urbański, M.: Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, vol. 148. Cambridge University Press, Cambridge (2003)26.Minkowski, H.: Geometrie der zahlen. Gesammelte Abhandlungen, vol. 2 (1911)27.Montgomery, H.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, vol. 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1994)28.Persson, T.: On a problem by R. Salem concerning minkowski’s question mark function (2015, preprint). arXiv:​1501.​00876 29.Pollington, A., Velani, S.: On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture. Acta Math. 185(2), 287–306 (2000)MathSciNetCrossRefMATH30.Queffélec, M., Ramaré, O.: Analyse de Fourier des fractions continues à quotients restreints. Enseign. Math. (2) 49(3–4), 335–356 (2003)31.Salem, R.: On some singular monotonic functions which are strictly increasing. Trans. Am. Math. Soc. 53, 427–439 (1943)MathSciNetCrossRefMATH32.Sarig, O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217(3), 555–577 (2001)MathSciNetCrossRefMATH33.Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Am. Math. Soc. 131(6), 1751–1758 (2003, electronic)34.Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Trans. Am. Math. Soc. 236, 121–153 (1978)MathSciNetCrossRefMATH35.Wolff, T.H.: Lectures on Harmonic Analysis, University Lecture Series, vol. 29. American Mathematical Society, Providence (2003) (with a foreword by Charles Fefferman and preface by Izabella Łaba, Edited by Łaba and Carol Shubin)36.Yakubovich, S.: The affirmitive solution to Salem’s problem revisited (2015, preprint). arXiv:​1501.​00141 37.Young, L.-S.: Large deviations in dynamical systems. Trans. Am. Math. Soc. 318(2), 525–543 (1990)MathSciNetMATH38.Yuri, M.: Large deviations for countable to one Markov systems. Commun. Math. Phys. 258(2), 455–474 (2005)MathSciNetCrossRefMATH About this Article Title Fourier transforms of Gibbs measures for the Gauss map Journal Mathematische Annalen Volume 364, Issue 3-4 , pp 983-1023 Cover Date2016-04 DOI 10.1007/s00208-015-1241-9 Print ISSN 0025-5831 Online ISSN 1432-1807 Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords 42A38 11K50 37C30 60F10 Industry Sectors Finance, Business & Banking Authors Thomas Jordan (1) Tuomas Sahlsten (2) Author Affiliations 1. School of Mathematics, University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, England, UK 2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel Continue reading... To view the rest of this content please follow the download PDF link above.