Complexity and fractal dimensions for infinite sequences with positive entropy

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the {\it word entropy} $E_W(f)$ associated to a given function $f$ and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by $f$ in terms of its word entropy. We present a combinatorial proof of the fact that $E_W(f)$ is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by $f$ and we give several examples showing that even under strong conditions on $f$, the word entropy $E_W(f)$ can be strictly smaller than the limiting lower exponential growth rate of $f$.Comment: 24 page

An algorithm for the word entropy

For any infinite word $w$ on a finite alphabet $A$, the complexity function $p_w$ of $w$ is the sequence counting, for each non-negative $n$, the number $p_w(n)$ of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$ and the the entropy of $w$ is the quantity $E(w)=\lim\limits_{n\to\infty}\frac 1n\log p_w(n)$. For any given function $f$ with exponential growth, Mauduit and Moreira introduced in [MM17] the notion of word entropy $E_W(f) = \sup \{E(w), w \in A^{{\mathbb N}}, p_w \le f \}$ and showed its links with fractal dimensions of sets of infinite sequences with complexity function bounded by $f$. The goal of this work is to give an algorithm to estimate with arbitrary precision $E_W(f)$ from finitely many values of $f$

ON SARNAK'S CONJECTURE AND VEECH'S QUESTION FOR INTERVAL EXCHANGES

International audienceUsing a criterion due to Bourgain [10] and the generalization of the self-dual induction defined in [18], for each primitive permutation we build a large family of k-interval exchanges satisfying Sarnak's conjecture, and, for at least one permutation in each Rauzy class, smaller families for which we have weak mixing, which implies a prime number theorem, and simplicity in the sense of Veech

Generation of further pseudorandom binary sequences, I (Blowing up a single sequence)

Assume that a binary sequence is given with strong pseudorandom properties. An algorithm is presented and studied which prepares many further binary sequences from the given one. It is shown that if certain conditions hold then each of the sequences obtained in this way also possesses strong pseudorandom properties. Moreover, it is proved that certain large families of these sequences also posses strong pseudorandom properties

On reducible and primitive subsets of FpF_p, I

A set $A\subset \mathbb F_p$ is said to be reducible if it can be represented in the form $A=B+C$ with $B, C\subset\mathbb F_p$, $|B|, |C|\ge 2$. If there are no sets $B, C$ with these properties then $A$ is said to be primitive. First three criteria are presented for primitivity of subsets of $F_p$. Then the distance between a given set $A\subset\mathbb F_p$ and the closest primitive set is studied

On linear complexity of binary lattices

The linear complexity is an important and frequently used measure of unpredictability and pseudorandomness of binary sequences. In this paper our goal is to extend this notion to two dimensions. We will define and study the linear complexity of binary lattices. The linear complexity of a truly random binary lattice will be estimated. Finally, we will analyze the connection between the linear complexity and the correlation measures, and we will utilize the inequalities obtained in this way for estimating the linear complexity of an important special binary lattice. Finally, we will study the connection between the linear complexity of binary lattices and of the associated binary sequences

On finite pseudorandom binary lattices

Pseudorandom binary sequences play a crucial role in cryptography. The classical approach to pseudorandomness of binary sequences is based on computational complexity. This approach has certain weak points thus in the last two decades years a new, more constructive and quantitative approach has been developed. Since multidimensional analogs of binary sequences (called binary lattices) also have important applications thus it is a natural idea to extend this new approach to the multidimensional case. This extension started with a paper published in 2006, and since that about 25 papers have been written on this subject. Here our goal is to present a survey of all these papers

The cross-correlation measure for families of binary sequences

Large families of binary sequences of the same length are considered and a new measure, the cross-correlation measure of order $k$ is introduced to study the connection between the sequences belonging to the family. It is shown that this new measure is related to certain other important properties of families of binary sequences. Then the size of the cross-correlation measure is studied. Finally, the cross-correlation measures of two important families of pseudorandom binary sequences are estimated