Irrationality exponent, Hausdorff dimension and effectivization

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] is robust, but surprisingly, that the effective continued fraction dimension and effective (base- b ) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that effective dimension is invariant with respect to base conversion [ 8 ]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\) but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the “non-faithfulness” of certain families of covers [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α−p/q|<q −μ has infinitely many solutions (p,q)∈ℤ×ℕ. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n⩾1. We then use this result to prove that for b⩾2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.

The notion of normal numbers was first introduced by Emile Borel [3] in 1909. He considered the decimal expansion of real numbers in the unit interval to the base r and assuming that every digit of their decimal expansions is independent and also takes all possible values 0,1, ・ ・・ and r-1 with equal probability, he proved that almost all real numbers are normal to the base r in the sense of Lebesgue measure. For a real number w, we denote {w} the fractional part of w defined by

Let ξ \xi be a real irrational number. We are interested in sequences of linear forms in 1 and ξ \xi , with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of ξ \xi . We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin’s new proof of Nesterenko’s linear independence criterion.

A refinement of an ergodic theorem and its application to Hardy functions

Let $\beta\in(1,2)$ and $x\in [0,1/(\beta-1)]$. We call a sequence $(\epsilon_{i})_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ a $\beta$-expansion for $x$ if $x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}$. We call a finite sequence $(\epsilon_{i})_{i=1}^{n}\in\

Martin-Löf randomness and Galton–Watson processes

The members of Martin-Lof random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Lof random Galton-Watson trees with survival parameter $\frac{2}{3}$. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than $\gamma=\log_2 \frac{3}{2}$, and a necessary condition is to have effective Hausdorff dimension greater than or equal to *** .

Monotonous betting strategies in warped casinos

Let x∈(0,1) be an irrational number with continued fraction expansion [a1(x),a2(x),⋯,an(x),⋯] . We give the multifractal spectrum of the irrationality exponent and the convergence exponent of x defined by v(x):=sup{v>0:|x−pq|<1qv for infinitely many (q,p)∈ℕ×ℤ} and τx:=infs⩾0:∑n⩾1an−sx<∞ respectively. To be precise, we completely determine the Hausdorff dimension of Eα,v=x∈0,1: τx=α, vx=v for any α⩾0, v⩾2 .

We construct a sequence of measurable functions converging at each point of the unit interval, but the set of points with any given rate of convergence has Hausdorff dimension one. This is used to show that a version of Egoroff’s theorem due to Taylor is best possible. The construction relies on an analysis of the maximal run length of ones in the dyadic expansion of real numbers. It is also proved that the exceptional set for a limit theorem of Renyi has Hausdorff dimension one.

The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its properties has just begun, and understanding the topology and complexity of its invariant set is of interest. In this paper, we conduct a local analysis on the first return map associated to a sliding Shilnikov connection, which reveals a conformal iterated function system (CIFS) structure. By using the theory of CIFS, we estimate the Hausdorff dimension of the local invariant set of the first return map, showing, in particular, that is a positive number smaller than 1, and with one-dimensional Lebesgue measure equal to zero. Moreover, we prove that the closure of the local invariant set is a Cantor set and retains both the Hausdorff dimension and Lebesgue measure of the local invariant set. Furthermore, this closure consists of the local invariant set along with the set of all pre-images, under the first return map, of the visible fold-regular point contained in the connection.

A divergence formula for randomness and dimension

Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).

The iterative elimination of the middle spacing in the random division of intervals with two points ldquoat randomrdquo - in the narrow sense of uniformly distributed - generates a random middle Cantor set. We compute the Hausdorff dimension (which intuitively evaluates how ldquodenserdquo a set is) of the random middle third Cantor set, and we verify that although the deterministic middle third Cantor set is the expectation of the random middle third Cantor set, it is more dense than its stochastic counterpart. This can be explained by the dependence of order statistics.