Related papers: Periodicity in Rectangular Arrays

Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. A submonoid $M$ generated by $k$ elements of $A^*$ is…

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals, and then apply them to proving primitive recursiveness of some natural problems in linear algebra and analysis. In…

We develop the ratios conjecture with one shift in the numerator and denominator in certain ranges for families of primitive quadratic Hecke $L$-functions of imaginary quadratic number fields with class number one using multiple Dirichlet…

Let $m\neq n$. An $m\times n\times p$ proper array is a three-dimensional array composed of directed cubes that obeys certain constraints. Because of these constraints, the $m\times n\times p$ proper arrays may be classified via a schema in…

We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…

Our main goal is to study the freeness of Sch\"utzenberger groups defined by primitive substitutions. Our findings include a simple freeness test for these groups, which is applied to exhibit a primitive invertible substitution with…

In this short note, we first associate a new simple undirected graph with a given word over an ordered alphabet of $n$-letters. We will call it the Lyndon graph of that word. Then, we introduce the concept of the Lyndon-word representable…

Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem} asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9…

We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive…

The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose difference is $d=2$. Generalizing this idea, we define new similar product mappings,…

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words…

In the present paper we propose generalizations of the regularity and counting lemmas for multidimensional matrices under a finite alphabet. Firstly, we prove a variant of a multidimensional regularity lemma with the help of a translation…

When $p(t)$ is a polynomial of degree $d$, $k$-th column of the Riordan array $\bigl(1/(1 - t^{d+1}), tp(t)\bigr)$ is an eventually periodic sequence with the repeating part beginning at the $1 + (k-1)(d+1)$-st term. The pre-periodic terms…

We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "runs" conjecture (Kolpakov \& Kucherov (FOCS '99)), which states that the…

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for…

We count flags of primitive lattices, which are objects of the form ${0}=\Lambda^{(0)}<\Lambda^{(1)}< \cdots <\Lambda^{(\ell)}= \mathbb{Z}^n$, where every $\Lambda^{(i)}$ is a primitive lattice in $\mathbb{Z}^n$. The counting is with…

Schinzel's original problem was to describe when an expression f(x)-g(y), with f,g nonconstant and having complex coefficients, is reducible. We call such an (f,g) a Schinzel pair if this happens nontrivially: f(x)-g(y) is newly reducible.…

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…

We give an asymptotic expression for the number of nonsingular integer n-by-n-matrices with primitive row vectors, determinant k, and Euclidean matrix norm less than T, for large T. We also investigate the density of matrices with primitive…

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This…