Related papers: Gaps in the Thue--Morse word
In this paper several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the…
We give estimates for the first two moments of arithmetical sequences in progressions. Instead of using the standard approximation, we work with a generalization of Vaughan's major arcs approximation which is similar to that appearing in…
We provide a non-trivial measure of irrationality for a class of Mahler numbers defined with infinite products which cover the Thue-Morse constant.
We discuss the base 3/2 representation of the natural numbers. We prove that the sum of digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum of digits…
Infinite sequences are of tremendous theoretical and practical importance, and in the Information Age sequences of 0s and 1s are of particular interest. Over the past century, the field of symbolic dynamics has developed to study sequences…
In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…
In this note we classify sequences according to whether they are morphic, pure morphic, uniform morphic, pure uniform morphic, primitive morphic, or pure primitive morphic, and for each possibility we either give an example or prove that no…
We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin…
We study the length of the gaps between consecutive members in the sumset sA when A is a pseudo s-th power sequence, with s>1. We show that, almost surely, limsup (b_{n+1}-b_{n})/log (b_n) = s^s s!/\Gamma^s(1/s), where b_n are the elements…
We introduce an algorithm for detection of bugs in sequential circuits. This algorithm is incomplete i.e. its failure to find a bug breaking a property P does not imply that P holds. The appeal of incomplete algorithms is that they scale…
We study a process of generating random positive integer weight sequences $\{ W_n \}$ where the gaps between the weights $\{ X_n = W_n - W_{n-1} \}$ are i.i.d. positive integer-valued random variables. We show that as long as the gap…
We introduce an algorithm for the uniform generation of infinite traces, i.e., infinite words up to commutation of some letters. The algorithm outputs on-the-fly approximations of a theoretical infinite trace, the latter being distributed…
A substring $u$ of a string $T$ is called a minimal unique substring (MUS) of $T$ if $u$ occurs exactly once in $T$ and any proper substring of $u$ occurs at least twice in $T$. A string $w$ is called a minimal absent word (MAW) of $T$ if…
An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This…
An absent factor of a string $w$ is a string $u$ which does not occur as a contiguous substring (a.k.a. factor) inside $w$. We extend this well-studied notion and define absent subsequences: a string $u$ is an absent subsequence of a string…
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation…
We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric…
The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive…
A \emph{morphism} is a mapping that transforms words through letter-wise substitution, where each symbol is consistently replaced by a fixed word. In the field of combinatorics on words, one topic that has attracted considerable attention…