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References

Published online by Cambridge University Press:  19 January 2017

Christopher J. Bishop
Affiliation:
Stony Brook University, State University of New York
Yuval Peres
Affiliation:
Microsoft Research, Redmond, WA
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References

Adelman, Omer. 1985. Brownian motion never increases: a new proof to a result of Dvoretzky, Erdʺos and Kakutani. Israel J. Math., 50(3), 189–192.Google Scholar
Adelman, Omer, Burdzy, Krzysztof, and Pemantle, Robin. 1998. Sets avoided by Brownian motion. Ann. Probab., 26(2), 429–464.Google Scholar
Aizenman, M., and Burchard, A. 1999. Hölder regularity and dimension bounds for random curves. Duke Math. J., 99(3), 419–453.Google Scholar
Ala-Mattila, Vesa. 2011. Geometric characterizations for Patterson–Sullivan measures of geometrically finite Kleinian groups. Ann. Acad. Sci. Fenn. Math. Diss., 120. Dissertation, University of Helsinki, Helsinki, 2011.
Arora, Sanjeev. 1998. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM, 45(5), 753–782.Google Scholar
Arveson, William. 1976. An Invitation to C*-algebras. Springer-Verlag, New York–Heidelberg. Graduate Texts in Mathematics, No. 39.
Athreya Krishna, B., and Ney Peter, E. 1972. Branching Processes. New York: Springer-Verlag. Die Grundlehren der Mathematischen Wissenschaften, Band 196.
Avila, Artur, and Lyubich, Mikhail. 2008. Hausdorff dimension and conformal measures of Feigenbaum Julia sets. J. Amer. Math. Soc., 21(2), 305–363.Google Scholar
Azzam, Jonas. 2015. Hausdorff dimension of wiggly metric spaces. Ark. Mat., 53(1), 1–36.Google Scholar
Babichenko, Yakov, Peres, Yuval, Peretz, Ron, Sousi, Perla, and Winkler, Peter. 2014. Hunter, Cauchy rabbit, and optimal Kakeya sets. Trans. Amer. Math. Soc., 366(10), 5567–5586.Google Scholar
Bachelier, L. 1900. Théorie de la speculation. Ann. Sci. Ecole Norm. Sup., 17, 21–86.Google Scholar
Balka, Richárd, and Peres, Yuval. 2014. Restrictions of Brownian motion. C. R. Math. Acad. Sci. Paris, 352(12), 1057–1061.Google Scholar
Balka, Richárd, and Peres, Yuval. 2016. Uniform dimension results for fractional Brownian motion. preprint, arXiv:1509.02979 [math.PR].
Bandt, Christoph, and Graf, Siegfried. 1992. Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc., 114(4), 995–1001.Google Scholar
Barański, Krzysztof. 2007. Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math., 210(1), 215–245.Google Scholar
Barański, Krzysztof, Bárány, Balázs, and Romanowska, Julia. 2014. On the dimension of the graph of the classical Weierstrass function. Adv. Math., 265, 32–59.Google Scholar
Barlow Martin, T., and Perkins, Edwin. 1984. Levels at which every Brownian excursion is exceptional. Pages 1–28 of: Seminar on probability, XVIII. Lecture Notes in Math., vol. 1059. Berlin: Springer.
Barlow Martin, T., and Taylor S., James. 1992. Defining fractal subsets of Z d . Proc. London Math. Soc. (3), 64(1), 125–152.Google Scholar
Bass Richard, F. 1995. Probabilistic Techniques in Analysis. Probability and its Applications (New York). New York: Springer-Verlag.
Bass Richard, F., and Burdzy, Krzysztof. 1999. Cutting Brownian paths. Mem. Amer. Math. Soc., 137(657), x+95.Google Scholar
Bateman, Michael, and Katz Nets, Hawk. 2008. Kakeya sets in Cantor directions. Math. Res. Lett., 15(1), 73–81.Google Scholar
Bateman, Michael, and Volberg, Alexander. 2010. An estimate from below for the Buffon needle probability of the four-corner Cantor set. Math. Res. Lett., 17(5), 959–967.Google Scholar
Benjamini, Itai, and Peres, Yuval. 1992. Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist., 28(4), 557–592.Google Scholar
Benjamini, Itai, and Peres, Yuval. 1994. Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields, 98(1), 91–112.Google Scholar
Benjamini, Itai, Pemantle, Robin, and Peres, Yuval. 1995. Martin capacity for Markov chains. Ann. Probab., 23(3), 1332–1346.Google Scholar
Berman Simeon, M. 1983. Nonincrease almost everywhere of certain measurable functions with applications to stochastic processes. Proc. Amer. Math. Soc., 88(1), 141–144.Google Scholar
Bertoin, Jean. 1996. Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge: Cambridge University Press.
Bertrand-Mathis, Anne. 1986. Ensembles intersectifs et récurrence de Poincaré. Israel J. Math., 55(2), 184–198.Google Scholar
Besicovitch, A.S. 1919. Sur deux questions d'intégrabilité des fonctions. J. Soc. Phys.- Math. (Perm'), 2(1), 105–123.Google Scholar
Besicovitch, A.S. 1928. On Kakeya's problem and a similar one. Math. Z., 27(1), 312–320.Google Scholar
Besicovitch, A.S. 1935. On the sum of digits of real numbers represented in the dyadic system. Math. Ann., 110(1), 321–330.Google Scholar
Besicovitch, A.S. 1938a. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann., 115(1), 296–329.Google Scholar
Besicovitch, A.S. 1938b. On the fundamental geometrical properties of linearly measurable plane sets of points II. Math. Ann., 115, 296–329. Google Scholar
Besicovitch, A.S. 1952. On existence of subsets of finite measure of sets of infinite measure. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math., 14, 339–344.Google Scholar
Besicovitch, A.S. 1956. On the definition of tangents to sets of infinite linear measure. Proc. Cambridge Philos. Soc., 52, 20–29.Google Scholar
Besicovitch, A.S. 1964. On fundamental geometric properties of plane line-sets. J. London Math. Soc., 39, 441–448.Google Scholar
Besicovitch, A.S., and Moran, P.A.P. 1945. The measure of product and cylinder sets. J. London Math. Soc., 20, 110–120.Google Scholar
Besicovitch, A.S., and Taylor, S.J. 1954. On the complementary intervals of a linear closed set of zero Lebesgue measure. J. London Math. Soc., 29, 449–459.Google Scholar
Besicovitch, A.S., and Ursell, H.D. 1937. Sets of fractional dimension v: On dimensional numbers of some continuous curves. J. London Math. Soc., 12, 18–25.Google Scholar
Bickel Peter, J. 1967. Some contributions to the theory of order statistics. Pages 575– 591 of: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calf., 1965/66), Vol. I: Statistics. Berkeley, Calif.: Univ. California Press.
Billingsley, Patrick. 1961. Hausdorff dimension in probability theory. II. Illinois J. Math., 5, 291–298.Google Scholar
Binder, Ilia, and Braverman, Mark. 2009. The complexity of simulating Brownian motion. Pages 58–67 of: Proceedings of the Twentieth Annual ACM–SIAM Symposium on Discrete Algorithms. Philadelphia, PA: SIAM.
Binder, Ilia, and Braverman, Mark. 2012. The rate of convergence of the walk on spheres algorithm. Geom. Funct. Anal., 22(3), 558–587.Google Scholar
Bishop Christopher, J 1996. Minkowski dimension and the Poincaré exponent. Michigan Math. J., 43(2), 231–246.Google Scholar
Bishop Christopher, J 1997. Geometric exponents and Kleinian groups. Invent. Math., 127(1), 33–50.Google Scholar
Bishop Christopher, J 2001. Divergence groups have the Bowen property. Ann. of Math. (2), 154(1), 205–217.Google Scholar
Bishop Christopher, J 2002. Non-rectifiable limit sets of dimension one. Rev. Mat. Iberoamericana, 18(3), 653–684.Google Scholar
Bishop Christopher, J 2007. Conformal welding and Koebe's theorem. Ann. of Math. (2), 166(3), 613–656.Google Scholar
Bishop Christopher, J, and Jones Peter, W. 1990. Harmonic measure and arclength. Ann. of Math. (2), 132(3), 511–547.Google Scholar
Bishop Christopher, J, and Jones Peter, W. 1994a. Harmonic measure, L2 estimates and the Schwarzian derivative. J. Anal. Math., 62, 77–113.Google Scholar
Bishop Christopher, J, and Jones Peter, W. 1994b. Harmonic measure, L2 estimates and the Schwarzian derivative. J. D'Analyse Math., 62, 77–114.Google Scholar
Bishop Christopher, J, and Jones Peter, W. 1997. Hausdorff dimension and Kleinian groups. Acta Math., 179(1), 1–39.Google Scholar
Bishop Christopher, J, and Peres, Yuval. 1996. Packing dimension and Cartesian products. Trans. Amer. Math. Soc., 348(11), 4433–4445.Google Scholar
Bishop Christopher, J, and Steger, Tim. 1993. Representation-theoretic rigidity in PSL(2, R). Acta Math., 170(1), 121–149.Google Scholar
Bishop Christopher, J, Jones Peter, W., Pemantle, Robin, and Peres, Yuval. 1997. The dimension of the Brownian frontier is greater than 1. J. Funct. Anal., 143(2), 309–336.Google Scholar
Bonk, Mario. 2011. Uniformization of Sierpiński carpets in the plane. Invent. Math., 186(3), 559–665.Google Scholar
Bonk, Mario, and Merenkov, Sergei. 2013. Quasisymmetric rigidity of square Sierpiński carpets. Ann. of Math. (2), 177(2), 591–643.Google Scholar
Bonk, Mario, Kleiner, Bruce, and Merenkov, Sergei. 2009. Rigidity of Schottky sets. Amer. J. Math., 131(2), 409–443.Google Scholar
Bourgain, J. 1987. Ruzsa's problem on sets of recurrence. Israel J. Math., 59(2), 150– 166.Google Scholar
Bourgain, J. 1991. Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal., 1(2), 147–187.Google Scholar
Bourgain, J. 1999. On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal., 9(2), 256–282.Google Scholar
Boyd David, W. 1973. The residual set dimension of the Apollonian packing. Mathematika, 20, 170–174.Google Scholar
Brown, R. 1828. A brief description of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Ann. Phys., 14, 294–313.Google Scholar
Burdzy, Krzysztof. 1989. Cut points on Brownian paths. Ann. Probab., 17(3), 1012– 1036.Google Scholar
Burdzy, Krzysztof. 1990. On nonincrease of Brownian motion. Ann. Probab., 18(3), 978–980.Google Scholar
Burdzy, Krzysztof, and Lawler Gregory, F. 1990. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab., 18(3), 981–1009.Google Scholar
Cajar, Helmut. 1981. Billingsley Dimension in Probability Spaces. Lecture Notes in Mathematics, vol. 892. Berlin: Springer-Verlag.
Carleson, Lennart. 1967. Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, No. 13. D. Van Nostrand Co., Inc., Princeton N.J.–Toronto, Ont.–London.
Carleson, Lennart. 1980. The work of Charles Fefferman. Pages 53–56 of: Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Helsinki: Acad. Sci. Fennica.
Chang, S.-Y.A., Wilson, J.M., and Wolff, T.H. 1985. Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv., 60(2), 217–246.Google Scholar
Charmoy Philippe, H.A., Peres, Yuval, and Sousi, Perla. 2014. Minkowski dimension of Brownian motion with drift. J. Fractal Geom., 1(2), 153–176.Google Scholar
Chayes, J.T., Chayes, L., and Durrett, R. 1988. Connectivity properties of Mandelbrot's percolation process. Probab. Theory Related Fields, 77(3), 307–324.Google Scholar
Cheeger, Jeff, and Kleiner, Bruce. 2009. Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property. Geom. Funct. Anal., 19(4), 1017–1028.Google Scholar
Chow Yuan, Shih, and Teicher, Henry. 1997. Probability Theory. Third edn. Independence, Interchangeability, Martingales. Springer Texts in Statistics. New York: Springer-Verlag.
Christ, Michael. 1984. Estimates for th. k-plane transform. Indiana Univ. Math. J., 33(6), 891–910.Google Scholar
Christensen Jens Peter, Reus. 1972. On sets of Haar measure zero in abelian Polish groups. Pages 255–260 (1973) of: Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), vol. 13.
Ciesielski, Z., and Taylor, S.J. 1962. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc., 103, 434–450.Google Scholar
Colebrook, C.M. 1970. The Hausdorff dimension of certain sets of nonnormal numbers. Michigan Math. J., 17, 103–116.Google Scholar
Cόrdoba, Antonio. 1993. The fat needle problem. Bull. London Math. Soc., 25(1), 81–82.Google Scholar
Cover Thomas, M., and Thomas Joy, A. 1991. Elements of Information Theory. Wiley Series in Telecommunications. New York: John Wiley & Sons Inc.
Cunningham, Jr., F. 1971. The Kakeya problem for simply connected and for starshaped sets. Amer. Math. Monthly, 78, 114–129.Google Scholar
Cunningham, Jr., F. 1974. Three Kakeya problems. Amer. Math. Monthly, 81, 582–592.Google Scholar
David, Guy. 1984. Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm Sup., 17, 157–189.Google Scholar
David, Guy. 1998. Unrectifiable 1-sets have vanishing analytic capacity. Rev. Mat. Iberoamericana, 14(2), 369–479.Google Scholar
Davies Roy, O. 1952. On accessibility of plane sets and differentiation of functions of two real variables. Proc. Cambridge Philos. Soc., 48, 215–232.Google Scholar
Davies Roy, O. 1971. Some remarks on the Kakeya problem. Proc. Cambridge Philos. Soc., 69, 417–421.Google Scholar
de Leeuw, Karel. 1965. O. Lp multipliers. Ann. of Math. (2), 81, 364–379.Google Scholar
Dekking, F.M., and Grimmett, G.R. 1988. Superbranching processes and projections of random Cantor sets. Probab. Theory Related Fields, 78(3), 335–355.Google Scholar
Drury, S.W. 1983. Lp estimates for the X-ray transform. Illinois J. Math., 27(1), 125–129.
Dubins Lester, E. 1968. On a theorem of Skorohod. Ann. Math. Statist., 39, 2094–2097.Google Scholar
Dudley, R.M. 2002. Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge. Revised reprint of the 1989 original.
Dudziak James, J. 2010. Vitushkin's Conjecture for Removable Sets. Universitext. New York: Springer.
Duistermaat, J.J. 1991. Self-similarity of “Riemann's nondifferentiable function”. Nieuw Arch. Wisk. (4), 9(3), 303–337.Google Scholar
Duplantier, Bertrand. 2006. Brownian motion, “diverse and undulating”. Pages 201– 293 of: Einstein, 1905–2005. Prog. Math. Phys., vol. 47. Basel: Birkhäuser. Translated from the French by Emily Parks.
Durrett, Richard. 1996. Probability: Theory and Examples. Belmont, CA: Duxbury Press.
Dvir, Zeev. 2009. On the size of Kakeya sets in finite fields. J. Amer. Math. Soc., 22(4), 1093–1097.Google Scholar
Dvir, Zeev, and Wigderson, Avi. 2011. Kakeya sets, new mergers, and old extractors. SIAM J. Comput., 40(3), 778–792.Google Scholar
Dvir, Zeev, Kopparty, Swastik, Saraf, Shubhangi, and Sudan, Madhu. 2009. Extensions to the method of multiplicities, with applications to Kakeya sets and mergers. Pages 181–190 of: 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009). IEEE Computer Soc., Los Alamitos, CA.
Dvoretzky, A., Erdʺos, P., and Kakutani, S. 1950. Double points of paths of Brownian motion i. n-space. Acta Sci. Math. Szeged, 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B), 75–81.Google Scholar
Dvoretzky, A., Erdʺos, P., and Kakutani, S. 1961. Nonincrease everywhere of the Brownian motion process. Pages 103–116 of: Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II. Berkeley, Calif.: Univ. California Press.
Dynkin, E.B., and Yushkevich, A.A. 1956. Strong Markov processes. Theory Probab. Appl., 1, 134–139.Google Scholar
Edgar Gerald, A. (ed). 2004. Classics on Fractals. Studies in Nonlinearity. Westview Press. Advanced Book Program, Boulder, CO.
Edmonds, Jack. 1965. Paths, trees, and flowers. Canad. J. Math., 17, 449–467.Google Scholar
Eggleston, H.G. 1949. The fractional dimension of a set defined by decimal properties. Quart. J. Math., Oxford Ser., 20, 31–36.Google Scholar
Einstein, A. 1905. Über die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Physik, 17, 549–560.Google Scholar
Elekes, György, Kaplan, Haim, and Sharir, Micha. 2011. On lines, joints, and incidences in three dimensions. J. Combin. Theory Ser. A, 118(3), 962–977.Google Scholar
Elekes, Márton, and Steprāns, Juris. 2014. Haar null sets and the consistent reflection of non-meagreness. Canad. J. Math., 66(2), 303–322.Google Scholar
Elekes, Márton, and Vidnyánszky, Zoltán. 2015. Haar null sets without G™?hulls. Israel J. Math., 209(1), 199–214.Google Scholar
Ellenberg Jordan, S., Oberlin, Richard, and Tao, Terence. 2010. The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika, 56(1), 1–25.Google Scholar
Erdʺos, P. 1949. On a theorem of Hsu and Robbins. Ann. Math. Statistics, 20, 286–291.Google Scholar
Erdʺos, P. 1961. A problem about prime numbers and the random walk. II. Illinois J. Math., 5, 352–353.Google Scholar
Erdʺos, Paul. 1940. On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math., 62, 180–186.Google Scholar
Evans Steven, N. 1992. Polar and nonpolar sets for a tree indexed process. Ann. Probab., 20(2), 579–590.Google Scholar
Falconer, K.J. 1980. Continuity properties o. k-plane integrals and Besicovitch sets. Math. Proc. Cambridge Philos. Soc., 87(2), 221–226.Google Scholar
Falconer, K.J. 1982. Hausdorff dimension and the exceptional set of projections. Mathematika, 29(1), 109–115.Google Scholar
Falconer, K.J. 1985. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85. Cambridge: Cambridge University Press.
Falconer, K.J. 1986. Sets with prescribed projections and Nikodým sets. Proc. London Math. Soc. (3), 53(1), 48–64.Google Scholar
Falconer, K.J. 1988. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc., 103(2), 339–350.Google Scholar
Falconer, K.J. 1989a. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc., 106(2), 543–554.Google Scholar
Falconer, K.J. 1989b. Projections of random Cantor sets. J. Theoret. Probab., 2(1), 65–70.Google Scholar
Falconer, K.J. 1990. Fractal Geometry. Chichester: John Wiley & Sons Ltd.
Falconer, K.J. 2013. Dimensions of self-affine sets: a survey. Pages 115–134 of: Further Developments in Fractals and Related Fields. Trends Math. Birkhäuser/Springer, New York.
Falconer, K.J., and Howroyd, J.D. 1996. Projection theorems for box and packing dimensions. Math. Proc. Cambridge Philos. Soc., 119(2), 287–295.Google Scholar
Fang, X. 1990. The Cauchy integral of Calderόn and analytic capacity. Ph.D. thesis, Yale University.
Fefferman, Charles. 1971. The multiplier problem for the ball. Ann. of Math. (2), 94, 330–336.Google Scholar
Feller, William. 1966. An Introduction to Probability Theory and its Applications. Vol. II. New York: John Wiley & Sons Inc.
Ferguson, Andrew, Jordan, Thomas, and Shmerkin, Pablo. 2010. The Hausdorff dimension of the projections of self-affine carpets. Fund. Math., 209(3), 193–213.Google Scholar
Ferrari, Fausto, Franchi, Bruno, and Pajot, Hervé. 2007. The geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 23(2), 437–480.Google Scholar
Folland, Gerald B. 1999. Real Analysis. Pure and Applied Mathematics (New York). New York: John Wiley & Sons Inc.
Ford Jr., L.R., and Fulkerson, D.R. 1962. Flows in Networks. Princeton, N.J.: Princeton University Press.
Freedman, David. 1971. Brownian Motion and Diffusion. San Francisco, Calif.: Holden-Day.
Freud, Geza. 1962. Über trigonometrische approximation und Fouriersche reihen. Math. Z., 78, 252–262.Google Scholar
Frostman, O. 1935. Potential d'equilibre et capacité des ensembles avec quelques applications `a la théorie des fonctions. Meddel. Lunds Univ. Math. Sen., 3(1-118).Google Scholar
Fujiwara, M., and Kakeya, S. 1917. On some problems for the maxima and minima for the curve of constant breadth and the in-revolvable curve of the equilateral triangle. Tohoku Math. J., 11, 92–110.Google Scholar
Furstenberg, H. 1967. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49.Google Scholar
Furstenberg, H. 1970. Intersections of Cantor sets and transversality of semigroups. Pages 41–59 of: Problems in Analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969). Princeton Univ. Press, Princeton, N.J.
Furstenberg, H. 1981. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, N.J.: Princeton University Press.
Gabow, H.N. 1990. Data structures of weighted matching and nearest common ancestors with linking. In: Proceedings of the 1st Annual ACM–SIAM Symposium on Discrete Algorithms.
Garnett John, B. 1970. Positive length but zero analytic capacity. Proc. Amer. Math. Soc., 24, 696–699.Google Scholar
Garnett John, B. 1981. Bounded Analytic Functions. Academic Press.
Garnett John, B., and Marshall Donald, E. 2005. Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge: Cambridge University Press.
Gerver, Joseph. 1970. The differentiability of the Riemann function at certain rational multiples of?. Amer. J. Math., 92, 33–55.Google Scholar
Graczyk, Jacek, and Smirnov, Stanislav. 2009. Non-uniform hyperbolicity in complex dynamics. Invent. Math., 175(2), 335–415.Google Scholar
Graf, Siegfried, Mauldin R., Daniel, and Williams, S.C. 1988. The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc., 71(381), x+121.Google Scholar
Guth, Larry. 2010. The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture. Acta Math., 205(2), 263–286.Google Scholar
Hahlomaa, Immo. 2005. Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math., 185(2), 143–169.Google Scholar
Hahlomaa, Immo. 2007. Curvature integral and Lipschitz parametrization in 1-regular metric spaces. Ann. Acad. Sci. Fenn. Math., 32(1), 99–123.Google Scholar
Hahlomaa, Immo. 2008. Menger curvature and rectifiability in metric spaces. Adv. Math., 219(6), 1894–1915.Google Scholar
Hamilton David, H. 1995. Length of Julia curves. Pacific J. Math., 169(1), 75–93.Google Scholar
Har-Peled, Sariel. 2011. Geometric Approximation Algorithms. Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence, RI.
Hardy, G.H. 1916. Weierstrass's non-differentiable function. Trans. Amer. Math. Soc., 17(3), 301–325.Google Scholar
Harris, T.E. 1960. A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13–20.Google Scholar
Hartman, Philip, and Wintner, Aurel. 1941. On the law of the iterated logarithm. Amer. J. Math., 63, 169–176.Google Scholar
Hausdorff, Felix. 1918. Dimension und äußeres Maß. Math. Ann., 79(1-2), 157–179.Google Scholar
Hawkes, John. 1975. Some algebraic properties of small sets. Quart. J. Math. Oxford Ser. (2), 26(102), 195–201.Google Scholar
Hawkes, John. 1981. Trees generated by a simple branching process. J. London Math. Soc. (2), 24(2), 373–384.Google Scholar
Hedenmalm, Haken. 2015. Bloch functions, asymptotic variance and zero packing. preprint, arXiv:1602.03358 [math.CV].
Hewitt, Edwin, and Savage Leonard, J. 1955. Symmetric measures on Cartesian products. Trans. Amer. Math. Soc., 80, 470–501.Google Scholar
Hewitt, Edwin, and Stromberg, Karl. 1975. Real and Abstract Analysis. New York: Springer-Verlag.
Hochman, Michael. 2013. Dynamics on fractals and fractal distributions. arXiv:1008.3731v2.
Hochman, Michael. 2014. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2), 180(2), 773–822.Google Scholar
Hochman, Michael. 2015. On self-similar sets with overlaps and inverse theorems for entropy in Rd. In preparation.
Hochman, Michael, and Shmerkin, Pablo. 2012. Local entropy averages and projections of fractal measures. Ann. of Math. (2), 175(3), 1001–1059.Google Scholar
Housworth Elizabeth, Ann. 1994. Escape rate for 2-dimensional Brownian motion conditioned to be transient with application to Zygmund functions. Trans. Amer. Math. Soc., 343(2), 843–852.Google Scholar
Howroyd, J.D. 1995. On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. London Math. Soc. (3), 70(3), 581–604.Google Scholar
Hsu, P.L., and Robbins, Herbert. 1947. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A., 33, 25–31.Google Scholar
Hunt Brian, R. 1994. The prevalence of continuous nowhere differentiable functions. Proc. Amer. Math. Soc., 122(3), 711–717.Google Scholar
Hunt Brian, R. 1998. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc., 126(3), 791–800.Google Scholar
Hunt Brian, R, Sauer, Tim, and Yorke James, A. 1992. Prevalence: a translationinvariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.), 27(2), 217–238.Google Scholar
Hunt Brian, R, Sauer, Tim, and Yorke James, A. 1993. Prevalence. An addendum to: “Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces” [Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238; MR1161274 (93k:28018)]. Bull. Amer. Math. Soc. (N.S.), 28(2), 306–307.Google Scholar
Hunt, G.A. 1956. Some theorems concerning Brownian motion. Trans. Amer. Math. Soc., 81, 294–319.Google Scholar
Hutchinson John, E. 1981. Fractals and self-similarity. Indiana Univ. Math. J., 30(5), 713–747.Google Scholar
Ivrii, Oleg. 2016. Quasicircles of dimension 1 + k2 do not exist. Preprint, arXiv:1511.07240 [math.DS].
Izumi, Masako, Izumi, Shin-ichi, and Kahane, Jean-Pierre. 1965. Théor`emes élémentaires sur les séries de Fourier lacunaires. J. Analyse Math., 14, 235–246.Google Scholar
Jodeit Jr., Max. 1971. A note on Fourier multipliers. Proc. Amer. Math. Soc., 27, 423–424.Google Scholar
Jones Peter, W. 1989. Square functions, Cauchy integrals, analytic capacity, and harmonic measure. Pages 24–68 of: Harmonic Analysis and Partial Differential Equations (El Escorial, 1987). Lecture Notes in Math., vol. 1384. Berlin: Springer.
Jones Peter, W. 1990. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1), 1–15.Google Scholar
Jones Peter, W. 1991. The traveling salesman problem and harmonic analysis. Publ. Mat., 35(1), 259–267. Conference on Mathematical Analysis (El Escorial, 1989).Google Scholar
Jordan, Thomas, and Pollicott, Mark. 2006. Properties of measures supported on fat Sierpinski carpets. Ergodic Theory Dynam. Systems, 26(3), 739–754.Google Scholar
Joyce, H., and Preiss, D. 1995. On the existence of subsets of finite positive packing measure. Mathematika, 42(1), 15–24.Google Scholar
Juillet, Nicolas. 2010. A counterexample for the geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 26(3), 1035–1056.Google Scholar
Kahane, J.-P. 1964. Lacunary Taylor and Fourier series. Bull. Amer. Math. Soc., 70, 199–213.Google Scholar
Kahane, J.-P. 1969. Trois notes sure les ensembles parfait linéaires. Enseigement Math., 15, 185–192.Google Scholar
Kahane, J.-P. 1985. Some Random Series of Functions. Second edn. Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge: Cambridge University Press.
Kahane, J.-P., and Peyri`ere, J. 1976. Sur certaines martingales de Benoit Mandelbrot. Advances in Math., 22(2), 131–145.Google Scholar
Kahane, J.-P., Weiss, Mary, and Weiss, Guido. 1963. On lacunary power series. Ark. Mat., 5, 1–26 (1963).Google Scholar
Kakeya, S. 1917. Some problems on maxima and minima regarding ovals. Tohoku Science Reports, 6, 71–88.Google Scholar
Kakutani, Shizuo. 1944. Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo, 20, 706–714.Google Scholar
Kamae, T., and Mend`es France, M. 1978. van der Corput's difference theorem. Israel J. Math., 31(3-4), 335–342.Google Scholar
Karatzas, Ioannis, and Shreve Steven, E. 1991. Brownian Motion and Stochastic Calculus. Second edn. Graduate Texts in Mathematics, vol. 113. New York: Springer-Verlag.
Karpińska, Boguslawa. 1999. Hausdorff dimension of the hairs without endpoints for exp z. C. R. Acad. Sci. Paris Sér. I Math., 328(11), 1039–1044.Google Scholar
Katz Nets, Hawk, and Tao, Terence. 2002. New bounds for Kakeya problems. J. Anal. Math., 87, 231–263. Dedicated to the memory of Thomas H. Wolff.Google Scholar
Katznelson, Y. 2001. Chromatic numbers of Cayley graphs on Z and recurrence. Combinatorica, 21(2), 211–219.Google Scholar
Kaufman, R. 1968. On Hausdorff dimension of projections. Mathematika, 15, 153–155.Google Scholar
Kaufman, R. 1969. Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris Sér. A-B, 268, A727–A728.Google Scholar
Kaufman, R. 1972. Measures of Hausdorff-type, and Brownian motion. Mathematika, 19, 115–119.Google Scholar
Kechris Alexander, S. 1995. Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer-Verlag, New York.
Kenyon, Richard. 1997. Projecting the one-dimensional Sierpinski gasket. Israel J. Math., 97, 221–238.Google Scholar
Kenyon, Richard, and Peres, Yuval. 1991. Intersecting random translates of invariant Cantor sets. Invent. Math., 104(3), 601–629.Google Scholar
Kenyon, Richard, and Peres, Yuval. 1996. Hausdorff dimensions of sofic affineinvariant sets. Israel J. Math., 94, 157–178.Google Scholar
Khinchin, A.Y. 1924. Über einen Satz der Wahrscheinlichkeitrechnung. Fund. Mat., 6, 9–20.Google Scholar
Khinchin, A.Y. 1933. Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Springer-Verlag.
Khintchine, A. 1926. Über eine Klasse linearer diophantischer Approximationen. Rendiconti Circ. Math. Palermo, 50(2), 211–219.Google Scholar
Khoshnevisan, Davar. 1994. A discrete fractal in Z1 +. Proc. Amer. Math. Soc., 120(2), 577–584.Google Scholar
Kinney, J.R. 1968. A thin set of circles. Amer. Math. Monthly, 75(10), 1077–1081.Google Scholar
Knight Frank, B. 1981. Essentials of Brownian Motion and Diffusion. Mathematical Surveys, vol. 18. Providence, R.I.: American Mathematical Society.
Kochen, Simon, and Stone, Charles. 1964. A note on the Borel–Cantelli lemma. Illinois J. Math., 8, 248–251.Google Scholar
Kolmogorov, A. 1929. Über das Gesetz des iterierten Logarithmus. Mathematische Annalen, 101(1), 126–135.Google Scholar
Körner, T.W. 2003. Besicovitch via Baire. Studia Math., 158(1), 65–78.Google Scholar
Kruskal, Jr., Joseph, B. 1956. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc., 7, 48–50.Google Scholar
Kuipers, L., and Niederreiter, H. 1974. Uniform Distribution of Sequences. Wiley– Interscience [John Wiley & Sons], New York–London–Sydney. Pure and Applied Mathematics.
Łaba, Izabella. 2008. From harmonic analysis to arithmetic combinatorics. Bull. Amer. Math. Soc. (N.S.), 45(1), 77–115.Google Scholar
Lalley Steven, P., and Gatzouras, Dimitrios. 1992. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J., 41(2), 533–568.Google Scholar
Lamperti, John. 1963. Wiener's test and Markov chains. J. Math. Anal. Appl., 6, 58–66.Google Scholar
Larman, D.H. 1967. On the Besicovitch dimension of the residual set of arbitrary packed disks in the plane. J. London Math. Soc., 42, 292–302.Google Scholar
Lawler Gregory, F. 1991. Intersections of Random Walks. Probability and its Applications. Boston, MA: Birkhäuser Boston Inc.
Lawler Gregory, F. 1996. The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab., 1, no. 5, 29–47 (electronic).Google Scholar
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2001a. The dimension of the planar Brownian frontier is 4/3. Math. Res. Lett., 8(4), 401–411.Google Scholar
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2001b. Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187(2), 237–273.Google Scholar
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2001c. Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187(2), 275–308.Google Scholar
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2002. Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist., 38(1), 109–123.Google Scholar
Le Gall, J.-F. 1987. The exact Hausdorff measure of Brownian multiple points. Pages 107–137 of: Seminar on Stochastic Processes, 1986 (Charlottesville, VA., 1986). Progr. Probab. Statist., vol. 13. Birkhäuser Boston, Boston, MA.
Lehmann, E.L. 1966. Some concepts of dependence. Ann. Math. Statist., 37, 1137– 1153.Google Scholar
Lerman, Gilad. 2003. Quantifying curvelike structures of measures by using L2 Jones quantities. Comm. Pure Appl. Math., 56(9), 1294–1365.Google Scholar
Lévy, Paul. 1940. Le mouvement brownien plan. Amer. J. Math., 62, 487–550.Google Scholar
Lévy, Paul. 1948. Processus stochastiques et mouvement Brownien. Suivi d'une note de M. Lo`eve. Paris: Gauthier–Villars.
Lévy, Paul. 1953. La mesure de Hausdorff de la courbe du mouvement Brownien. Giorn. Ist. Ital. Attuari, 16, 1–37 (1954).Google Scholar
Lindenstrauss, Elon, and Varju, Peter P. 2014. Random walks in the group of Euclidean isometries and self-similar measures. arXiv:1405.4426.
Lyons, Russell. 1989. The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys., 125(2), 337–353.Google Scholar
Lyons, Russell. 1990. Random walks and percolation on trees. Ann. Probab., 18(3), 931–958.Google Scholar
Lyons, Russell. 1992. Random walks, capacity and percolation on trees. Ann. Probab., 20(4), 2043–2088.Google Scholar
Lyons, Russell, and Pemantle, Robin. 1992. Random walk in a random environment and first-passage percolation on trees. Ann. Probab., 20(1), 125–136.Google Scholar
Lyons, Russell, and Peres, Yuval. 2016. Probability on Trees and Networks. Cambridge University Press.
Makarov, N.G. 1989. Probability methods in the theory of conformal mappings. Algebra i Analiz, 1(1), 3–59.Google Scholar
Mandelbrot, Benoit. 1974. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mechanics, 62, 331–358.Google Scholar
Markowsky, Greg. 2011. On the expected exit time of planar Brownian motion from simply connected domains. Electron. Commun. Probab., 16, 652–663.Google Scholar
Marstrand, J.M. 1954. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3), 4, 257–302.Google Scholar
Marstrand, J.M. 1979. Packing planes in R3. Mathematika, 26(2), 180–183.(1980).Google Scholar
Mattila, Pertti. 1990. Orthogonal projections, Riesz capacities, and Minkowski content. Indiana Univ. Math. J., 39(1), 185–198.Google Scholar
Mattila, Pertti. 1995. Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge: Cambridge University Press. Fractals and rectifiability.
Mattila, Pertti. 2015. Fourier Analysis and Hausdorff Dimension. Cambridge Studies in Advanced Mathematics, vol. 150. Cambridge University Press.
Mattila, Pertti, and Mauldin, R. Daniel. 1997. Measure and dimension functions: measurability and densities. Math. Proc. Cambridge Philos. Soc., 121(1), 81–100.
Mattila, Pertti, and Vuorinen, Matti. 1990. Linear approximation property, Minkowski dimension, and quasiconformal spheres. J. London Math. Soc. (2), 42(2), 249– 266.Google Scholar
Mattila, Pertti, Melnikov Mark, S., and Verdera, Joan. 1996. The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. (2), 144(1), 127–136.Google Scholar
Mauldin R., Daniel, and Williams, S.C. 1986. On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc., 298(2), 793–803.Google Scholar
McKean Jr. Henry, P. 1961. A problem about prime numbers and the random walk. I. Illinois J. Math., 5, 351.Google Scholar
McKean Jr. Henry, P. 1955. Hausdorff–Besicovitch dimension of Brownian motion paths. Duke Math. J., 22, 229–234.Google Scholar
McMullen Curtis, T. 1984. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J., 96, 1–9.Google Scholar
McMullen Curtis, T. 1998. Hausdorff dimension and conformal dynamics. III. Computation of dimension. Amer. J. Math., 120(4), 691–721.Google Scholar
Melnikov Mark, S., and Verdera, Joan. 1995. A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs. Internat. Math. Res. Notices, 325–331.Google Scholar
Milnor, John. 2006. Dynamics in One Complex Variable. Third edn. Annals of Mathematics Studies, vol. 160. Princeton, NJ: Princeton University Press.
Mitchell Joseph, S.B. 1999. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k- MST, and related problems. SIAM J. Comput., 28(4), 1298–1309.(electronic).Google Scholar
Mitchell Joseph, S.B. 2004. Shortest paths and networks. Chap. 27, pages 607–641 of: Goodman, Jacob E., and O'Rourke, Joseph (eds), Handbook of Discrete and Computational Geometry (2nd Edition). Boca Raton, FL: Chapman & Hall/CRC.
Montgomery Hugh, L. 2001. Harmonic analysis as found in analytic number theory. Pages 271–293 of: Twentieth Century Harmonic Analysis – a Celebration (Il Ciocco, 2000). NATO Sci. Ser. II Math. Phys. Chem., vol. 33. Kluwer Acad. Publ., Dordrecht.
Moran, P.A.P. 1946. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc., 42, 15–23.Google Scholar
Mörters, Peter, and Peres, Yuval. 2010. Brownian Motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press.
Muller Mervin, E. 1956. Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Statist., 27, 569–589.Google Scholar
Nazarov, F., Peres, Y., and Volberg, A. 2010. The power law for the Buffon needle probability of the four-corner Cantor set. Algebra i Analiz, 22(1), 82–97.Google Scholar
Nevanlinna, Rolf. 1936. Eindeutige Analytische Funktionen. Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, vol. 46. J. Springer, Berlin.
Newhouse Sheldon, E. 1970. Nondensity of axiom A(a) on S2. Pages 191–202 of: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968). Amer. Math. Soc., Providence, R.I.
Nikodym, O. 1927. Sur la measure des ensembles plan dont tous les points sont rectalineair ément accessibles. Fund. Math., 10, 116–168.Google Scholar
Oberlin, Richard. 2010. Two bounds for the X-ray transform. Math. Z., 266(3), 623–644.Google Scholar
Oh, Hee. 2014. Apollonian circle packings: dynamics and number theory. Jpn. J. Math., 9(1), 69–97.Google Scholar
Okikiolu, Kate. 1992. Characterization of subsets of rectifiable curves in R n. J. London Math. Soc. (2), 46(2), 336–348.
O'Neill Michael, D. 2000. Anderson's conjecture for domains with fractal boundary. Rocky Mountain J. Math., 30(1), 341–352.Google Scholar
Pajot, Hervé. 2002. Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Mathematics, vol. 1799. Berlin: Springer-Verlag.
Pál, Julius. 1921. Ein minimumproblem für ovale. Math. Ann., 83(3-4), 311–319.Google Scholar
Paley, R.E.A.C., Wiener, N., and Zygmund, A. 1933. Notes on random functions. Math. Z., 37(1), 647–668.Google Scholar
Parry, William. 1964. Intrinsic Markov chains. Trans. Amer. Math. Soc., 112, 55–66.Google Scholar
Pemantle, Robin. 1997. The probability that Brownian motion almost contains a line. Ann. Inst. H. Poincaré Probab. Statist., 33(2), 147–165.Google Scholar
Pemantle, Robin, and Peres, Yuval. 1994. Domination between trees and application to an explosion problem. Ann. Probab., 22(1), 180–194.Google Scholar
Pemantle, Robin, and Peres, Yuval. 1995. Galton–Watson trees with the same mean have the same polar sets. Ann. Probab., 23(3), 1102–1124.Google Scholar
Peres, Yuval. 1994a. The packing measure of self-affine carpets. Math. Proc. Cambridge Philos. Soc., 115(3), 437–450.
Peres, Yuval. 1994b. The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Cambridge Philos. Soc., 116(3), 513–526.
Peres, Yuval. 1996a. Points of increase for random walks. Israel J. Math., 95, 341–347.
Peres, Yuval. 1996b. Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. H. Poincaré Phys. Théor., 64(3), 339–347.
Peres, Yuval, and Schlag, Wilhelm. 2000. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J., 102(2), 193–251.Google Scholar
Peres, Yuval, and Schlag, Wilhelm. 2010. Two Erdʺos problems on lacunary sequences: chromatic number and Diophantine approximation. Bull. Lond. Math. Soc., 42(2), 295–300.Google Scholar
Peres, Yuval, and Shmerkin, Pablo. 2009. Resonance between Cantor sets. Ergodic Theory Dynam. Systems, 29(1), 201–221.Google Scholar
Peres, Yuval, and Solomyak, Boris. 1996. Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett., 3(2), 231–239.Google Scholar
Peres, Yuval, and Solomyak, Boris. 2000. Problems on self-similar sets and self-affine sets: an update. Pages 95–106 of: Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998). Progr. Probab., vol. 46. Birkhäuser, Basel.
Peres, Yuval, and Solomyak, Boris. 2002. How likely is Buffon's needle to fall near a planar Cantor set. Pacific J. Math., 204(2), 473–496.Google Scholar
Perron, Oskar. 1929. Ü ber stabilität und asymptotisches verhalten der integrale von differentialgleichungssystemen. Math. Z., 29(1), 129–160.Google Scholar
Prause, István, and Smirnov, Stanislav. 2011. Quasisymmetric distortion spectrum. Bull. Lond. Math. Soc., 43(2), 267–277.Google Scholar
Przytycki, F., and Urbański, M. 1989. On the Hausdorff dimension of some fractal sets. Studia Math., 93(2), 155–186.Google Scholar
Quilodrán, René. 2009/10. The joints problem in Rn. SIAM J. Discrete Math., 23(4), 2211–2213.
Ray, Daniel. 1963. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc., 106, 436–444.Google Scholar
Revuz, Daniel, and Yor, Marc. 1994. Continuous Martingales and Brownian motion. Second edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293. Berlin: Springer-Verlag.
Rippon, P.J., and Stallard, G.M. 2005. Dimensions of Julia sets of meromorphic functions. J. London Math. Soc. (2), 71(3), 669–683.Google Scholar
Rogers, C.A., and Taylor, S.J. 1959. The analysis of additive set functions in Euclidean space. Acta Math., 101, 273–302.Google Scholar
Rohde, S. 1991. On conformal welding and quasicircles. Michigan Math. J., 38(1), 111–116.Google Scholar
Root, D.H. 1969. The existence of certain stopping times on Brownian motion. Ann. Math. Statist., 40, 715–718.Google Scholar
Rudin, Walter. 1987. Real and Complex Analysis. Third edn. New York: McGraw-Hill Book Co.
Ruzsa, I.Z., and Székely, G.J. 1982. Intersections of traces of random walks with fixed sets. Ann. Probab., 10(1), 132–136.Google Scholar
Saint Raymond, Xavier, and Tricot, Claude. 1988. Packing regularity of sets in n-space. Math. Proc. Cambridge Philos. Soc., 103(1), 133–145.Google Scholar
Salem, R., and Zygmund, A. 1945. Lacunary power series and Peano curves. Duke Math. J., 12, 569–578.Google Scholar
Saraf, Shubhangi, and Sudan, Madhu. 2008. An improved lower bound on the size of Kakeya sets over finite fields. Anal. PDE, 1(3), 375–379.Google Scholar
Sawyer, Eric. 1987. Families of plane curves having translates in a set of measure zero. Mathematika, 34(1), 69–76.Google Scholar
Schief, Andreas. 1994. Separation properties for self-similar sets. Proc. Amer. Math. Soc., 122(1), 111–115.Google Scholar
Schleicher, Dierk. 2007. The dynamical fine structure of iterated cosine maps and a dimension paradox. Duke Math. J., 136(2), 343–356.Google Scholar
Schoenberg, I.J. 1962. On the Besicovitch–Perron solution of the Kakeya problem. Pages 359–363 of: Studies in Mathematical Analysis and Related Topics. Stanford, Calif.: Stanford Univ. Press.
Schul, Raanan. 2005. Subsets of rectifiable curves in Hilbert space – the analyst's TSP. Ph.D. thesis, Yale University.
Schul, Raanan. 2007a. Ahlfors-regular curves in metric spaces. Ann. Acad. Sci. Fenn. Math., 32(2), 437–460.Google Scholar
Schul, Raanan. 2007b. Analyst's traveling salesman theorems. A survey. Pages 209–220 of: In the Tradition of Ahlfors–Bers. IV. Contemp. Math., vol. 432. Providence, RI: Amer. Math. Soc.
Schul, Raanan. 2007c. Subsets of rectifiable curves in Hilbert space – the analyst's TSP. J. Anal. Math., 103, 331–375.
Schwartz, J.T. 1980. Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach., 27(4), 701–717.Google Scholar
Selberg, Atle. 1952. The general sieve-method and its place in prime number theory. Pages 286–292 of: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1. Providence, R. I.: Amer. Math. Soc.
Shannon, C.E. 1948. A mathematical theory of communication. Bell System Tech. J., 27, 379–423, 623–656.Google Scholar
Shen, Weixiao. 2015. Hausdorff dimension of the graphs of the classical Weierstrass functions. Preprint, arXiv:1505.03986 [math.DS].
Shmerkin, Pablo. 2014. On the exceptional set for absolute continuity of Bernoulli convolutions. Geom. Funct. Anal., 24(3), 946–958.Google Scholar
Shmerkin, Pablo, and Solomyak, Boris. 2014. Absolute continuity of self-similar measures, their projections and convolutions. arXiv:1406.0204.
Skorokhod, A.V. 1965. Studies in the Theory of Random Processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass.
Smirnov, Stanislav. 2010. Dimension of quasicircles. Acta Math., 205(1), 189–197.Google Scholar
Solecki, Sławomir. 1996. On Haar null sets. Fund. Math., 149(3), 205–210.Google Scholar
Solomyak, Boris. 1995. On the random series © ± n(an Erdʺos problem). Ann. of Math. (2), 142(3), 611–625.Google Scholar
Solomyak, Boris. 1997. On the measure of arithmetic sums of Cantor sets. Indag. Math. (N.S.), 8(1), 133–141.Google Scholar
Solomyak, Boris, and Xu, Hui. 2003. On the ‘Mandelbrot set’ for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity, 16(5), 1733–1749.Google Scholar
Spitzer, Frank. 1964. Principles of Random Walk. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London.
Strassen, V. 1964. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3, 211–226 (1964).Google Scholar
Stratmann Bernd, O. 2004. The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones. Pages 93–107 of: Fractal Geometry and Stochastics III. Progr. Probab., vol. 57. Basel: Birkhäuser.
Sullivan, Dennis. 1982. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math., 149(3-4), 215–237.Google Scholar
Sullivan, Dennis. 1984. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153(3-4), 259–277.Google Scholar
Talagrand, Michel, and Xiao, Yimin. 1996. Fractional Brownian motion and packing dimension. J. Theoret. Probab., 9(3), 579–593.Google Scholar
Taylor S., James. 1953. The Hausdorff 〈-dimensional measure of Brownian paths in n-space. Proc. Cambridge Philos. Soc., 49, 31–39.Google Scholar
Taylor S., James. 1964. The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Philos. Soc., 60, 253–258.Google Scholar
Taylor S., James. 1966. Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 5, 247–264.Google Scholar
Taylor S., James, and Tricot, Claude. 1985. Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc., 288(2), 679–699.
Taylor S., James, and Wendel, J.G. 1966. The exact Hausdorff measure of the zero set of a stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 6, 170–180.Google Scholar
Tolsa, Xavier. 2003. Painlevé's problem and the semiadditivity of analytic capacity. Acta Math., 190(1), 105–149.Google Scholar
Tolsa, Xavier. 2014. Analytic Capacity, the Cauchy Transform, and Nonhomogeneous Calderόn–Zygmund theory. Progress in Mathematics, vol. 307. Birkhäuser/Springer, Cham.
Tricot, Claude. 1981. Douze definitions de la densité logarithmique. Comptes Rendus Acad. Sci. Paris, 293, 549–552.Google Scholar
Tricot, Claude. 1984. A new proof for the residual set dimension of the Apollonian packing. Math. Proc. Cambridge Philos. Soc., 96(3), 413–423.Google Scholar
Tricot, Claude. 1982. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc., 91(1), 57–74.Google Scholar
Tukia, Pekka. 1989. Hausdorff dimension and quasisymmetric mappings. Math. Scand., 65(1), 152–160.Google Scholar
Urbański, Mariusz. 1990. The Hausdorff dimension of the graphs of continuous selfaffine functions. Proc. Amer. Math. Soc., 108(4), 921–930.Google Scholar
Urbański, Mariusz. 1991. On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point. Studia Math., 97(3), 167–188.Google Scholar
Urbański, Mariusz. 1997. Geometry and ergodic theory of conformal non-recurrent dynamics. Ergodic Theory Dynam. Systems, 17(6), 1449–1476.Google Scholar
van der Corput, J.G. 1931. Diophantische ungleichungen. I. Zur gleichverteilung modulo eins. Acta Math., 56(1), 373–456.Google Scholar
Volkmann, Bodo. 1958. Ü ber Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind. VI. Math. Z., 68, 439–449.Google Scholar
Weierstrass, K. 1872. Ü ber contiuirliche functionen eines reellen arguments, die für keinen werth des letzeren einen bestimmten differentialquotienten besitzen. Königl. Akad. Wiss., 3, 71–74. Mathematische Werke II.Google Scholar
Wiener, N. 1923. Differential space. J. Math. Phys., 2(6), 1319–1362.Google Scholar
Wolff, Thomas. 1995. An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana, 11(3), 651–674.Google Scholar
Wolff, Thomas. 1999. Recent work connected with the Kakeya problem. Pages 129–162 of: Prospects in Mathematics (Princeton, NJ, 1996). Providence, RI: Amer. Math. Soc.
Xiao, Yimin. 1996. Packing dimension, Hausdorff dimension and Cartesian product sets. Math. Proc. Cambridge Philos. Soc., 120(3), 535–546.Google Scholar
Zdunik, Anna. 1990. Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math., 99(3), 627–649.Google Scholar
Zhan, Dapeng. 2011. Loop-erasure of planar Brownian motion. Comm. Math. Phys., 303(3), 709–720.Google Scholar
Zippel, Richard. 1979. Probabilistic algorithms for sparse polynomials. Pages 216– 226 of: Symbolic and Algebraic Computation (EUROSAM ‘79, Internat. Sympos., Marseille, 1979). Lecture Notes in Comput. Sci., vol. 72. Berlin: Springer.
Zygmund, A. 1959. Trigonometric Series. Cambridge University Press.

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  • References
  • Christopher J. Bishop, Stony Brook University, State University of New York, Yuval Peres
  • Book: Fractals in Probability and Analysis
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