We regard the classic Thue–Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of the local measure decay (superpolynomial). For several scaling functions, we show that points with this behavior are abundant in the sense of full Hausdorff dimension. At the fastest possible scaling, the corresponding rates reveal several remarkable phenomena. There is a gap between level sets for dyadic rationals and non-dyadic points, and beyond dyadic rationals, non-zero accumulation points occur only within intervals of positive length. The dependence between the smallest and the largest accumulation point also manifests itself in a non-trivial joint dimension spectrum.

We consider the generalized Thue-Morse sequences (t_n^(c))_n≥ 0 (c ∈ [0,1) being a parameter) defined by t_n^(c) = e^2π i c s_2(n), where s_2(n) is the sum of digits of the binary expansion of n. For the polynomials σ_N^(c) (x) := ∑_n=0^N-1 t_n^(c) e^2π i n x, we have proved in [18] that the uniform norm σ_N^(c)_∞ behaves like N^γ(c) and the best exponent γ(c) is computed. In this paper, we study the pointwise behavior and give a complete multifractal analysis of the limit lim_n→∞n^-1log |σ_2^n^(c)(x)|.

Given an integer \begin{document}$ q\ge 2 $\end{document} and a real number \begin{document}$ c\in [0,1) $\end{document}, consider the generalized Thue-Morse sequence \begin{document}$ (t_n^{(q;c)})_{n\ge 0} $\end{document} defined by \begin{document}$ t_n^{(q;c)} = e^{2\pi i c s_q(n)} $\end{document}, where \begin{document}$ s_q(n) $\end{document} is the sum of digits of the \begin{document}$ q $\end{document}-expansion of \begin{document}$ n $\end{document}. We prove that the \begin{document}$ L^\infty $\end{document}-norm of the trigonometric polynomials \begin{document}$ \sigma_{N}^{(q;c)} (x) : = \sum_{n = 0}^{N-1} t_n^{(q;c)} e^{2\pi i n x} $\end{document}, behaves like \begin{document}$ N^{\gamma(q;c)} $\end{document}, where \begin{document}$ \gamma(q;c) $\end{document} is equal to the dynamical maximal value of \begin{document}$ \log_q \left|\frac{\sin q\pi (x+c)}{\sin \pi (x+c)}\right| $\end{document} relative to the dynamics \begin{document}$ x \mapsto qx \mod 1 $\end{document} and that the maximum value is attained by a \begin{document}$ q $\end{document}-Sturmian measure. Numerical values of \begin{document}$ \gamma(q;c) $\end{document} can be computed.

This article is about various aspects of the Fourier dimension and its variants. One aspect is to relate, and contrast the Fourier dimension with the Hausdorff dimension. Moreover we will present some questions where the Fourier dimension can be successfully applied. This includes uniform distribution problems and questions from geometric measure theory like the occurrence of Salem sets. There have been several similar but different notions of the Fourier dimension subject to different applications. We will argue that these various notions are indeed different and also do not behave like a regular dimension-like quantity. We also will give an alternative more regular definition that still reflects most of the important properties that are needed for applications.

For a given number alpha is an element of (0, 1) and a 1-periodic function f, we study the convergence of the series Sigma(infinity)(n=1) f(x+n alpha)/n, called one-sided Hilbert transform relative to the rotation x bar right arrow x + alpha mod 1. Among others, we prove that for any non-polynomial function of class C-2 having Taylor-Fourier series (i.e. Fourier coefficients vanish on Z_), there exists an irrational number alpha (actually a residual set of alpha) such that the series diverges for all x. We also prove that for any irrational number alpha, there exists a continuous function f such that the series diverges for all x. The convergence of general series Sigma(infinity)(n=1) a(n)f(x + n alpha) is also discussed in different cases involving the diophantine property of the number alpha and the regularity of the function f.

Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant measures invariant under linear maps of the circle.

We study Markov interval maps with random holes. The holes are not necessarily elements of the Markov partition. Under a suitable, and physically relevant, assumption on the noise, we show that the transfer operator associated with the random open system can be reduced to a transfer operator associated with the closed deterministic system. Exploiting this fact, we show that the random open system admits a unique (meaningful) absolutely continuous conditionally stationary measure. Moreover, we prove the existence of a unique probability equilibrium measure supported on the survival set, and we study its Hausdorff dimension.

We study the asymptotics of the scenery flow. We give corrected versions with proofs of a certain lemma by Hochman, and study some related phenomena.

We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that a measure has modified Fourier dimension greater than or equal to $s$ if and only if it annihilates all sets with modified Fourier dimension less than $s$.

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σ m ={0, ...,m−1}ℕ that are invariant under multiplication by integers. The results apply to the sets {x∈Σ m :∀ k, x k x 2k ... x nk =0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

We provide a general mechanism for obtaining uniform information frompointwise data. For instance, a diffeomorphism of a compact Riemannianmanifold with pointwise expanding and contracting continuous invariant conefamilies is an Anosov diffeomorphism, i.e., the entire manifold is uniformlyhyperbolic.

Let (X, T) be a topological dynamical system and let Φ: X^r →ℝ be a continuous function on the product space X^r= X× ... × X (r≥ 1). We are interested in the limit of V-statistics taking Φ as kernel: [lim_n→∞ n^-r∑_1≤i_1, ..., i_r≤n Φ(T^i_1x, ..., T^i_r x).] The multifractal spectrum of topological entropy of the above limit is expressed by a variational principle when the system satisfies the specification property. Unlike the classical case (r=1) where the spectrum is an analytic function when Φ is Hölder continuous, the spectrum of the limit of higher order V-statistics (r≥ 2) may be discontinuous even for very nice kernel Φ.

Let mu be a Gibbs measure of the doubling map T of the circle. For a mu-generic point x and a given sequence {r(n)} subset of R+, consider the intervals (T-n x - r(n) (mod 1), T-n x + r(n) (mod 1)). In analogy to the classical Dvoretzky covering of the circle, we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures that are multifractal. Such a principle was proved by Beresnevich and Velani [Ann. Math. 164 (2006) 971-992] for monofractal measures. In the symbolic language, we completely describe the combinatorial structure of a typical relatively short sequence; in particular, we can describe the occurrence of 'atypical' relatively long words. Our results have a direct and deep number-theoretical interpretation via inhomogeneous dyadic Diophantine approximation by numbers belonging to a given (dyadic) Diophantine class.

Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = A+...+A (k times). We show that for any non-decreasing sequence {a_k} taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension a_k for all k. We also show how to control various kinds of dimension simultaneously for families of iterated sumsets. These results are in stark contrast to the Plunnecke-Rusza inequalities in additive combinatorics. However, for lower box-counting dimension, the analogue of the Plunnecke-Rusza inequalities does hold.

We consider expansions of real numbers in non-integer bases. These expansions are generated by beta-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.

In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.

We consider approximations of real numbers by rational numbers with denominator 2 n .We will exploit results on hitting times for the underlying dynamical system on the full shift.In the second part we transfer the results to the β-shifts.This will give us an estimate on the approximation speed of arbitrary β-shifts by finite type β-shifts.This is a particular case of Katok's horseshoe approximation of non-uniformly hyperbolic systems.

Let $\mu$ be a Gibbs measure of the doubling map $T$ of the circle. For a $\mu$-generic point $x$ and a given sequence $\{r_n\} \subset \R^+$, consider the intervals $(T^nx - r_n \pmod 1, T^nx + r_n \pmod 1)$. In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures which are multifractal. Such a principle was shown by Beresnevich and Velani \cite{BV} only for monofractal measures. In the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence, in particular we can describe the occurrence of ''atypical'' relatively long words. Our results have a direct and deep number-theoretical interpretation via inhomogeneous diadic diophantine approximation by numbers belonging to a given (diadic) diophantine class.