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It is known that a real symmetric circulant matrix with diagonal entries d≥ 0, off-diagonal entries ±1 and orthogonal rows exists only of order 2d+ 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with ddifferent from an odd integer is n= 2d+ 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤm. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.

A word udefined over an alphabet $\mathcal A$is c-balanced (c∈$\mathbb N$) if for all pairs of factors v, wof uof the same length and for all letters a∈$\mathcal A$, the difference between the number of letters ain vand wis less or equal to c. In this paper we consider a ternary alphabet $\mathcal A$= {L, S, M} and a class of substitutions $\varphi_p$defined by $\varphi_p$(L) = LpS, $\varphi_p$(S) = M, $\varphi_p$(M) = Lp–1Swhere p>1. We prove that the fixed point of $\varphi_p$, formally written as $\varphi_p^\infty$(L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e.they cannot be improved.

We study some arithmetical and combinatorial properties of β-integers for βbeing the larger root of the equation x2= mx - n,m,n ∈ ℵ, m ≥ n +2 ≥ 3. We determine with the accuracy of ±1 the maximal number of β-fractional positions, which may arise as a result of addition of two β-integers. For the infinite word uβ>coding distances between the consecutive β-integers, we determine precisely also the balance. The word uβ>is the only fixed point of the morphism A→Am-1Band B→Am-n-1B. In the case n = 1, the corresponding infinite word uβ>is sturmian, and, therefore, 1-balanced. On the simplest non-sturmian example with n≥2, we illustrate how closely the balance and the arithmetical properties of β-integers are related.

In this paper we will deal with the balance properties of the infinite binary words associated to β-integers when βis a quadratic simple Pisot number. Those words are the fixed points of the morphisms of the type $\varphi(A)=A^pB$, $\varphi(B)=A^q$for $p\in\mathbb N$, $q\in\mathbb N$, $p\geq q$, where $\beta=\frac{p+\sqrt{p^2+4q}}{2}$. We will prove that such word is t-balanced with $t=1+\left[(p-1)/(p+1-q)\right]$. Finally, in the case that p < qit is known [B. Adamczewski, Theoret. Comput. Sci.273(2002) 197–224] that the fixed point of the substitution $\varphi(A)=A^pB$, $\varphi(B)=A^q$is not m-balanced for any m. We exhibit an infinite sequence of pairs of words with the unbalance property.