Egoroff’s theorem and maximal run length

For any [Formula: see text], let [Formula: see text] be the partial summation of the first [Formula: see text] digits in the binary expansion of [Formula: see text] and [Formula: see text] be its run-length function. The classical Borel’s normal number theorem tells us that for almost all [Formula: see text], the limit of [Formula: see text] as [Formula: see text] goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] for almost every [Formula: see text] in [Formula: see text]. In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any [Formula: see text] and [Formula: see text], we completely determine the Hausdorff dimension of the following set: [Formula: see text] where [Formula: see text] and [Formula: see text] After some minor modifications, our result still holds if we replace the denominator [Formula: see text] in [Formula: see text] with any increasing function [Formula: see text] satisfying [Formula: see text] tending to [Formula: see text] and [Formula: see text]. As a result, we also obtain that the set of points for which neither the sequence [Formula: see text] nor [Formula: see text] converges has full Hausdorff dimension.

In this paper, we study some exceptional sets of points whose partial quotients in their Sylvester continued fraction expansions obey some restrictions. More precisely, for α ≥ 1 we prove that the Hausdorff dimension of the set [Formula: see text] is one. In addition, we find that the points whose partial quotients in their Sylvester continued fraction expansions obey some property of divisibility have the same Engel continued fraction expansion and Sylvester continued fraction expansion. And we establish that the set of points whose Engel continued fraction expansion and Sylvester continued fraction expansion coincide is uncountable.

On the range of simple symmetric random walks on the line

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

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

Pseudochaotic dynamics near global periodicity

In this paper we study normal subgroups of Kleinian groups as well as discrepancy groups (d-groups), that are Kleinian groups for which the exponent of convergence is strictly less than the Hausdorff dimension of the limit set. We show that the limit set of a d-group always contains a range of fractal subsets, each containing the set of radial limit points and having Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. We then consider normal subgroups $G$ of an arbitrary non-elementary Kleinian group $H$, and show that the exponent of convergence of $G$ is bounded from below by half of the exponent of convergene of $H$. Finally, we give a discussion of various examples of d-groups.

On the Chebyshev-Cramér Asymptotic Expansions

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

Infinite derivatives of the Takagi-Van der Waerden functions

The function s(u) arising in the study of long primitive BCH codes over GF(q) is reviewed. The set of points 0 < u ≤ 1 such that qku has a modulo 1 representation in the interval [a, 1] for every integer k ≥ 0 is shown to have Hausdorff dimension s(a) for every 0 ≤ a ≤ 1. Berlekamp's conjecture on the dimension of a set of points at which s fails to be differentiable is also proved.

§0. We begin by stating a theorem of Jarnik [1]. Let m be a positive integer, 0 0. Let E be the set of points (x1, …, xm) for which the inequalityadmits infinitely many solutions q ≥ 1. Then, supposingJarnik proves that E has infinite Hausdorff β-measure.

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

Fix an integer s. Let \(f:[0,1)^s \rightarrow {\mathbb {R}}\) be an integrable function. Let \({\mathscr {P}}\subset [0, 1]^s\) be a finite point set. Quasi-Monte Carlo integration of f by \({\mathscr {P}}\) is the average value of f over \({\mathscr {P}}\) that approximates the integration of f over the s-dimensional cube. Koksma–Hlawka inequality tells that, by a smart choice of \({\mathscr {P}}\), one may expect that the error decreases roughly \(O(N^{-1}(\log N)^s)\). For any \(\alpha \ge 1\), J. Dick gave a construction of point sets such that for \(\alpha \)-smooth f, convergence rate \(O(N^{-\alpha }(\log N)^{s\alpha })\) is assured. As a coarse version of his theory, M-Saito-Matoba introduced Walsh figure of Merit (WAFOM), which gives the convergence rate \(O(N^{-C\log N/s})\). WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of order \(N^{-\alpha }\) with \(\alpha >1\), for several test integrands for \(s=4\) and 8.

We show that on any translation surface, if a regular point is contained in some simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in $${\mathbb {RP}}^1$$. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist on some translation surfaces. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa’s classifications of periodic points and of $$\text {GL}(2,{\mathbb {R}})$$ orbit closures in hyperelliptic components, as well as a recent result of Eskin–Filip–Wright.