We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ
and expansive matrix M: Λ → Λ if ρ(M−1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M−1. We shall prove further that if the polynomial f(x) = c0 + c1x + ··· + ckxk ∈ Z[x], ck = 1 satisfies the condition |c0| > 2 Σ
|ci| then there is a suitable digit set D for which (Zk, M, D) is a number system, where M is the companion matrix of f(x).
We study the set of the representable numbers in base with ρ>1 and n∊ℕ and with digits in an arbitrary finite real alphabet A. We give a geometrical description of the convex hull of the representable numbers in base q and alphabet A and an explicit characterization of its extremal points. A characterizing condition for the convexity of the set of representable numbers is also shown.
Contrary to the classical situation, in noninteger bases almost all numbers have a continuum of distinct expansions. However, the set of numbers having a unique expansions also has a rich topological and combinatorial structure. We clarify the connection of this set with the sets of numbers having a unique infinite or doubly infinite expansion.
Authors:Shigeki Akiyama, Horst Brunotte, and Attila Pethő
The concept of a canonical number system can be regarded as a natural generalization of decimal representations of rational
integers to elements of residue class rings of polynomial rings. Generators of canonical number systems are CNS polynomials
which are known in the linear and quadratic cases, but whose complete description is still open. In the present note reducible
CNS polynomials are treated, and the main result is the characterization of reducible cubic CNS polynomials.
Authors:Z. Kobayashi, T. Kuzumaki, T. Okada, T. Sekiguchi, and Y. Shiota
We define a probability measure which has Markov property on the unit interval, compute a higher order partial derivative
of its distribution function about the parameter. As application, we obtain explicit formulas of a digital sum of the block
of the binary number.
We estimate multiplicative character sums over the integers with a fixed sum of binary digits and apply these results to study
the distribution of products of such integers in residues modulo a prime p. Such products have recently appeared in some cryptographic
algorithms, thus our results give some quantitative assurances of their pseudorandomness which is crucial for the security
of these algorithms.
Les propriétés multiplicatives des nombres ellipséphiques peuvent être obtenues à l’aide des moments de la série génératrice
de cette suite. Nous donnons des estimations précises pour les grands moments par deux méthodes distinctes: l’une combinatoire
fournit un résultat précis dans le cas réputé le plus difficile des nombres n’utilisant que les 0 et les 1; la seconde purement
analytique fournit un résultat sans condition sur les chiffres.
We investigate linear relations between pattern sequences in a 〈q,r〉-numeration system, and give a basis of the module generated by pattern sequences for words of length not exceeding l. The expressions of pattern sequences using the basis are also studied. Similar results are obtained for the module generated by all pattern sequences.