scholarly journals Hausdorff dimension of level sets of generalized Takagi functions

2014 ◽  
Vol 157 (2) ◽  
pp. 253-278
Author(s):  
PIETER C. ALLAART
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Continuous Functions ◽  
Zero Set ◽  
Arbitrary Integer ◽  
Open Problems ◽  
Formula Group ◽  
Nowhere Differentiable Function ◽  
Image Position ◽  
Class Of Functions

AbstractThis paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form \begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*} where φ(x) is the distance from x to the nearest integer, and for each n, ωn is a {−1,1}-valued function which is constant on each interval [j/2n,(j+1)/2n), j=0,1,. . .,2n − 1. This class of functions includes Takagi's continuous but nowhere differentiable function. It is shown that the largest possible Hausdorff dimension of f−1(y) is $\log ((9+\sqrt{105})/2)/\log 16\approx .8166$, but in case each ωn is constant, the largest possible dimension is 1/2. These results are extended to the intersection of the graph of f with lines of arbitrary integer slope. Furthermore, two natural models of choosing the signs ωn(x) at random are considered, and almost-sure results are obtained for the Hausdorff dimension of the zero set and the set of maximum points of f. The paper ends with a list of open problems.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS

2020 ◽  
Vol 102 (2) ◽  
pp. 186-195
Author(s):  
WEILIANG WANG ◽  
LU LI
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Continuous Functions ◽  
Lipschitz Functions ◽  
Image Position

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set $$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$ where $\unicode[STIX]{x1D70F}_{1}$, $\unicode[STIX]{x1D70F}_{2}$ are two positive continuous functions with $\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$ for all $x,y\in [0,1]$.

scholarly journals GENERALIZED HARDY–CESÀRO OPERATORS BETWEEN WEIGHTED SPACES

2018 ◽  
Vol 61 (1) ◽  
pp. 13-24
Author(s):  
THOMAS VILS PEDERSEN
Keyword(s):  
Bounded Operator ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Cesàro Operator ◽  
Weakly Compact ◽  
Locally Finite ◽  
Borel Measures ◽  
Formula Group ◽  
Image Position ◽  
Cesaro Operator

AbstractWe characterize those non-negative, measurable functions ψ on [0, 1] and positive, continuous functions ω1 and ω2 on ℝ+ for which the generalized Hardy–Cesàro operator $$\begin{equation*}(U_{\psi}f)(x)=\int_0^1 f(tx)\psi(t)\,dt\end{equation*}$$ defines a bounded operator Uψ: L1(ω1) → L1(ω2) This generalizes a result of Xiao [7] to weighted spaces. Furthermore, we extend Uψ to a bounded operator on M(ω1) with range in L1(ω2) ⊕ ℂδ0, where M(ω1) is the weighted space of locally finite, complex Borel measures on ℝ+. Finally, we show that the zero operator is the only weakly compact generalized Hardy–Cesàro operator from L1(ω1) to L1(ω2).

scholarly journals Topological Hausdorff dimension and level sets of generic continuous functions on fractals

2012 ◽  
Vol 45 (12) ◽  
pp. 1579-1589 ◽  
Author(s):  
Richárd Balka ◽  
Zoltán Buczolich ◽  
Márton Elekes
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Continuous Functions

scholarly journals The two-well problem with surface energy

2006 ◽  
Vol 136 (4) ◽  
pp. 795-805 ◽  
Author(s):  
Andrew Lorent
Keyword(s):  
Surface Energy ◽  
Lower Bounds ◽  
Continuous Functions ◽  
Convex Integration ◽  
Piecewise Affine ◽  
Bounded Lipschitz Domain ◽  
Affine Functions ◽  
Piecewise Affine Functions ◽  
Image Position ◽  
Class Of Functions

Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω > 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

On the Decomposition of A Class of Functions of Bounded Variation

1964 ◽  
Vol 16 ◽  
pp. 479-484 ◽  
Author(s):  
R. G. Laha
Keyword(s):  
Distribution Function ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Functions Of Bounded Variation ◽  
Image Position ◽  
Class Of Functions

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so thatthe above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,

scholarly journals Subexponentially increasing sums of partial quotients in continued fraction expansions

2015 ◽  
Vol 160 (3) ◽  
pp. 401-412 ◽  
Author(s):  
LINGMIN LIAO ◽  
MICHAŁ RAMS
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Point Of View ◽  
Partial Quotient ◽  
Formula One ◽  
Formula Group ◽  
Partial Quotients ◽  
Image Position ◽  
Continued Fraction Expansions ◽  
Increasing Function

AbstractWe investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S_{n}(x)=\sum_{j=1}^n a_{j}(x)$, where x = [a1(x), a2(x), . . .] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : $\mathbb{N}$ → $\mathbb{N}$, one is interested in the Hausdorff dimension of the set E_\varphi = \left\{x\in (0,1): \lim_{n\to\infty} \frac {S_n(x)} {\varphi(n)} =1\right\}. Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(nγ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), Eϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ(n) = exp(n1/2). In a similar way, the distribution of the largest partial quotient is also studied.

scholarly journals On primes not dividing binomial coefficients

1993 ◽  
Vol 113 (2) ◽  
pp. 225-232 ◽  
Author(s):  
J. W. Sander
Keyword(s):  
Binomial Coefficients ◽  
Open Problems ◽  
Image Position

AbstractWe prove thatthus dealing with open problems concerning divisors of binomial coefficients.

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

Sign in / Sign up

Export Citation Format

Share Document