In this paper we extend Champernowne’s construction of normal numbers in base to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base .
Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers and respectively, such that and are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.
Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.
In 2000, Florian Luca proved that F₁₀ = 55 and L₅ = 11 are the largest numbers with only one distinct digit in the Fibonacci and Lucas sequences, respectively. In this paper, we find terms of a linear recurrence sequence with only one block of digits in its expansion in base g ≥ 2. As an application, we generalize Luca's result by finding the Fibonacci and Lucas numbers with only one distinct block of digits of length up to 10 in its decimal expansion.
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real has algebraic degree , then the number of 1-bits in the expansion of through bit position satisfiesfor a positive number (depending on ) and sufficiently large . This in itself establishes the transcendency of a class of reals where the integer-valued...
Given an integer base and a completely -additive arithmetic function taking integer values, we deduce an asymptotic expression for the counting functionunder a mild restriction on the values of . When , the base sum of digits function, the integers counted by are the so-called base Niven numbers, and our result provides a generalization of the asymptotic known in that case.