On evolving chains of cube-free binary sequences

Chains of concatenated finite binary words are considered, where each word, except possibly the very first one, is composed of alternating blocks of zeroes and ones with block lengths not exceeding two. These chains are formed following two evolution schemes. The first scheme is standard, where alternating blocks are visited at random. In the second approach, proposed by us in this paper, each subsequent word of the chain is uniquely determined by its immediate predecessor, being formed as a specifically inflated version of that word. Famous Kolakoski sequence is then just one, very special example of such deterministic evolution when one starts from its third element. We present heuristic arguments supported by simulations indicating that all such deterministic infinite chains should have the asymptotic density of digit 1 equal 1/2 and that the subsequent word lengths asymptotically scale with factor of 3/2 and hence the density of 1’s in subsequent finite words may also tend to 1/2.

On Kakeya Conditions for Achievement Sets

We prove that for each infinite subset C of $${\mathbb {N}}$$ N there exists a sequence $$(x_n)$$ ( x n ) such that $$\{n: x_n>r_n\}=C$$ { n : x n > r n } = C and the achievement set $$A(x_n)$$ A ( x n ) is a Cantor set. Moreover, we show that it is possible to construct a sequence $$(x_n)$$ ( x n ) such that the set $$\{n: x_n>r_n\}$$ { n : x n > r n } has asymptotic density $$\alpha $$ α for each $$\alpha \in [0,1)$$ α ∈ [ 0 , 1 ) and $$A(x_n)$$ A ( x n ) is a Cantorval.

Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology

In this paper, the concept of probabilistic g-metric space with degree l, which is a generalization of probabilistic G-metric space, is introduced. Then, by endowing strong topology, the definition of l-dimensional asymptotic density of a subset A of $\mathbb{N}^{l}$ N l is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.

Density of sets with missing differences and applications

For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

Trees in Positive Entropy Subshifts

I give a simple proof for the fact that positive entropy subshifts contain infinite binary trees where branching happens synchronously in each branch, and that the branching times form a set with positive lower asymptotic density.

Kolakoski sequence: links between recurrence, symmetry and limit density

The Kolakoski sequence $S$ is the unique element of \(\left\lbrace 1,2 \right\rbrace^{\omega}\) starting with 1 and coinciding with its own run length encoding. We use the parity of the lengths of particular subclasses of initial words of \(S\) as a unifying tool to address the links between the main open questions - recurrence, mirror/reversal invariance and asymptotic density of digits. In particular we prove that recurrence implies reversal invariance, and give sufficient conditions which would imply that the density of 1s is \(\frac{1}{2}\).

A lower bound for Cusick’s conjecture on the digits of n + t

Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$ T. W. Cusick conjectured that c t > 1/2. We have the elementary bound 0 < c t < 1; however, no bound of the form 0 < α ≤ c t or c t ≤ β < 1, valid for all t, is known. In this paper, we prove that c t > 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).

Entropy-Based Classification of Elementary Cellular Automata under Asynchronous Updating: An Experimental Study

Classification of asynchronous elementary cellular automata (AECAs) was explored in the first place by Fates et al. (Complex Systems, 2004) who employed the asymptotic density of cells as a key metric to measure their robustness to stochastic transitions. Unfortunately, the asymptotic density seems unable to distinguish the robustnesses of all AECAs. In this paper, we put forward a method that goes one step further via adopting a metric entropy (Martin, Complex Systems, 2000), with the aim of measuring the asymptotic mean entropy of local pattern distribution in the cell space of any AECA. Numerical experiments demonstrate that such an entropy-based measure can actually facilitate a complete classification of the robustnesses of all AECA models, even when all local patterns are restricted to length 1. To gain more insights into the complexity concerning the forward evolution of all AECAs, we consider another entropy defined in the form of Kolmogorov–Sinai entropy and conduct preliminary experiments on classifying their uncertainties measured in terms of the proposed entropy. The results reveal that AECAs with low uncertainty tend to converge remarkably faster than models with high uncertainty.