TITLE

Ternary expansions of powers of 2

PUB. DATE
June 2009
SOURCE
Journal of the London Mathematical Society;Jun2009, Vol. 79 Issue 3, p562
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
Erdo˝s asked how frequently 2n has a ternary expansion that omits the digit 2. He conjectured that this holds only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first considers truncated ternary expansions of real sequences xn (λ) = ⌊λ2n ⌋, where λ > 0 is a real number, along with its untruncated version, whereas the second considers 3-adic expansions of sequences yn(λ) = λ2n, where λ is a 3-adic integer. We show in both cases that the set of initial values having infinitely many iterates that omit the digit 2 is small in a suitable sense. For each nonzero initial value we obtain an asymptotic upper bound as k → ∞ on the number of the first k iterates that omit the digit 2. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3-adic Cantor sets.
ACCESSION #
39988789

 

Related Articles

  • HARNACK'S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH. Toivanen, Olli // Annales Academiae Scientiarum Fennicae. Mathematica;2012, Vol. 37 Issue 2, p571 

    We prove Harnack's inequality for general solutions of elliptic equations -divA(x, u;∇u) = B(x, u,∇u), where A and B satisfy natural structural conditions with respect to a variable growth exponent p(x). The proof is based on a modification of the Caccioppoli inequality, which...

  • Strong convergence of an iterative method for pseudo-contractive and monotone mappings. Zegeye, Habtu; Shahzad, Naseer // Journal of Global Optimization;Sep2012, Vol. 54 Issue 1, p173 

    In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common...

  • ON INTERSECTIONS OF CANTOR SETS: SELF-SIMILARITY. PEDERSEN, STEEN; PHILLIPS, JASON D. // Communications in Mathematical Analysis;2014, Vol. 15 Issue 2, p1 

    Let C be a Cantor subset of the real line. For a real number t, let C+t be the translate of C by t. We say two real numbers s,t are translation equivalent, if the intersection of C and C+s is a translate of the intersection of C and C+t. We consider a class of Cantor sets determined by...

  • Expansion of weighted pseudoinverse matrices with singular weights into matrix power products and iteration methods. Sergienko, I.; Galba, E.; Deineka, V. // Ukrainian Mathematical Journal;Sep2007, Vol. 59 Issue 9, p1417 

    We obtain expansions of weighted pseudoinverse matrices with singular weights into matrix power products with negative exponents and arbitrary positive parameters. We show that the rate of convergence of these expansions depends on a parameter. On the basis of the proposed expansions, we...

  • Two-grid Interpolation Algorithms for Difference Schemes of Exponential Type for Semilinear Diffusion Convection-Dominated Equations. Vulkov, L. G.; Zadorin, A. I. // AIP Conference Proceedings;10/30/2008, Vol. 1067 Issue 1, p284 

    In this paper we propose two-grid algorithms for implementation of the A.M.Il’in’s scheme to diffusion convection-dominated equations. To find the solution from nonlinear algebraic systems we investigate Newton and Picard iterative methods. We offer to use the difference scheme on...

  • An FPGA Based High Speed IEEE - 754 Double Precision Floating Point Adder/Subtractor and Multiplier Using Verilog. Addanki, Purna Ramesh; Tilak Alapati, Venkata Nagaratna; Avana, Mallikarjuna Prasad // International Journal of Advanced Science & Technology;Mar2013, Vol. 52, p61 

    Floating Point (FP) addition, subtraction and multiplication are widely used in large set of scientific and signal processing computation. A high speed floating point double precision adder/subtractor and multiplier are implemented on a Virtex-6 FPGA. In addition, the proposed designs are...

  • Approaching the Power Logarithmic and Difference Means by Iterative Algorithms Involving the Power Binomial Mean. Raïssouli, Mustapha // International Journal of Mathematics & Mathematical Sciences;2011, p1 

    Introducing the notion of cross means we give iterative algorithms involving the power binomial mean and converging to the power logarithmic and difference means. At the end, we address a list of open problems derived from our present work.

  • The Favard length of product Cantor sets. Łaba, Izabella; Zhai, Kelan // Bulletin of the London Mathematical Society;Dec2010, Vol. 42 Issue 6, p997 

    Nazarov, Peres and Volberg proved recently that the Favard length of the nth iteration of the 4-corner Cantor set is bounded from above by n−c for an appropriate c. We extend this result to more general product Cantor sets whose projection in some direction has positive 1-dimensional measure.

  • An Exploration Of The Generalized Cantor Set. Islam Khan, Md. Shariful; Islam, Md. Shahidul // International Journal of Scientific & Technology Research;Jul2013, Vol. 2 Issue 7, p50 

    In this paper, we study the prototype of fractal of the classical Cantor middle-third set which consists of points along a line segment, and possesses a number of fascinating properties. We discuss the construction and the self-similarity of the Cantor set. We also generalized the construction...

Share

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics