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Some Problems and Ideas of Erdős in Analysis and Geometry

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Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

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Abstract

I review a meager few of the many problems and ideas Erdős proposed over the years involving a mixture of measure theory, geometry, and set theory.

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References

  1. J. Bourgain, On the Erdős-Volkmann and Katz-Tao Ring Conjectures, Geom. Funct. Anal., 19 (2003), 334–365.

    Article  MathSciNet  Google Scholar 

  2. Z. Buczolich and R. D. Mauldin, On the convergence of series of translates for measurable functions, Mathematika, 46 (1999), 337–341.

    Article  MATH  MathSciNet  Google Scholar 

  3. Z. Buczolich, J.-P. Kahane and R. D. Mauldin, On series of translates of positive functions, Acta Math. Hungarica, 98 (2001), 171–188.

    Article  MathSciNet  Google Scholar 

  4. Z. Buczolich and R. D. Mauldin, On series of translates of positive functions II, Indagationes Math., 12 (2001), 317–327.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. T. Croft, K. J. Falconer, Richard K. Guy, Unsolved problems in geometry, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  6. R. G. Downey and D. R. Hirschfeldt, Algorithmic randomness and complexity, Theory and Applications of Computability, Springer, New York, 2010.

    Google Scholar 

  7. G. A. Edgar and C. Miller, Borel subrings of the reals, Proc. Amer. Math. Soc., 131, 1121–1129, 2002.

    Article  MathSciNet  Google Scholar 

  8. P. Elias, Dirichlet sets, Erdős-Kunen-Mauldin theorem and analytic subgroups of the reals. Proc. Amer. Math. Soc., 139, (2010), 2093–2104.

    Article  MathSciNet  Google Scholar 

  9. P. Erdős, On the difference of consecutive primes, Quart. J. Oxford, 6 (1935), 124–128.

    Article  Google Scholar 

  10. P. Erdős, On the strong law of large numbers, Trans. Amer. Math. Soc., 67 (1949), 51–56.

    MathSciNet  Google Scholar 

  11. P. Erdős, Some results on additive number theory, Proc. Amer. Math. Soc., 5 (1954), 847–853.

    MathSciNet  Google Scholar 

  12. P. Erdős, Some remarks on set theory. IV, Mich Math. J., 2 (1953-54), 169–173 (1955).

    Article  Google Scholar 

  13. P. Erdős and B. Volkmann, Additive Gruppen mit vorgegebener Hausdorffscher dimension. J. Reine Angew. Math., 221 (1966), 203–208.

    MathSciNet  Google Scholar 

  14. P. Erdős, Set-theoretic, measure-theoretic, combinatorial and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange, 4, no. 2, (1978/79), 113–138.

    MathSciNet  Google Scholar 

  15. P. Erdős, My Scottish Book Problems, in: The Scottish Book, Mathematics from the Scottish Café. Edited by R. Daniel Mauldin, Birkhäuser, Boston, Mass., 1981.

    Google Scholar 

  16. P. Erdős, K. Kunen, R. D. Mauldin, Some additive properties of sets of real numbers, Fund. Math., 113 (1981), 187–199.

    MathSciNet  Google Scholar 

  17. P. Erdős, Some combinatorial, geometric and set theoretic problems in measure theory, in Measure Theory, Oberwolfach 1983, Lecture Notes in Mathematics 1089, Springer-Verlag (1984).

    Google Scholar 

  18. P. Erdős, S. Jackson, R. D. Mauldin, On partitions of lines and planes, Fund. Math., 145 (1994), 101–119.

    MathSciNet  Google Scholar 

  19. K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge university press, Cambridge, 1986.

    Google Scholar 

  20. K. Falconer, Fractal Geometry, John Wiley & Sons Inc., Hoboken, NJ, second edition, 2003.

    Book  MATH  Google Scholar 

  21. P. D. Humke, M. Laczkovich, Transference of Density, preprint, 2012.

    Google Scholar 

  22. D. Khoshnevisan, Probability, Graduate Studies in Mathematics, AMS, 2007.

    Google Scholar 

  23. L. A. Levin, The concept of a random sequence, Dokl. Akad. Nauk SSSR, 212 (1973), 548–550.

    MathSciNet  Google Scholar 

  24. Ming Li and Paul Vitanyi, An introduction to Kolmogorov complexity and its applications, third ed., Texts in Computer Science, Springer, New York, 2008.

    Google Scholar 

  25. G. G. Lorentz, On a problem of additive number theory, Proc. Amer. Math. Soc., 5 (1954), 838–841.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. H. Lutz, The dimensions of individual strings and sequences, Inform. and Comput., 187 (2003), no. 1, 49–79.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Laczkovich and I. Z. Rusza, Measure of sumsets and ejective sets I., Real Analysis Exchange, 22 (1996-1997), 153–166.

    MathSciNet  Google Scholar 

  28. R. D. Mauldin, Some problems in set theory, analysis and geometry, in Paul Erdős and his Mathematics I, 493–505, Springer, 2002.

    Google Scholar 

  29. R. D. Mauldin and M. Urbanski, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995–5025.

    Article  MATH  MathSciNet  Google Scholar 

  30. Elvira Mayordomo, A Kolmogorov complexity characterization of constructive Hausdorff dimension, Inform. Process. Lett., 84 (2002), no. 1, 1–3.

    Article  MATH  MathSciNet  Google Scholar 

  31. I. Z. Ruzsa, On a problem of P. Erdős, Canad. Math. Bull., 15 (1972), 309–310.

    Article  MATH  MathSciNet  Google Scholar 

  32. I. Z. Ruzsa, Additive completion of lacunary sequences, Combinatorics, 21 (2001), 279–291.

    Article  MATH  MathSciNet  Google Scholar 

  33. R. E. Svetic, The Erdős similarity problem, Real Analysis Exchange, 26(2) (2000), 525–540.

    MathSciNet  Google Scholar 

  34. Roger Webster, Convexity, Oxford University Press, 1994.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of North Texas, Denton, TX, 76203, USA

    R. Daniel Mauldin

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  1. R. Daniel Mauldin
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

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© 2013 János Bolyai Mathematical Society and Springer-Verlag

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Mauldin, R.D. (2013). Some Problems and Ideas of Erdős in Analysis and Geometry. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_13

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