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Finite and periodic orbits of shift radix systems

Peter Kirschenhofer, Attila Pethő, Paul Surer, Jörg Thuswaldner (2010)

Journal de Théorie des Nombres de Bordeaux

For r = ( r 0 , ... , r d - 1 ) ∈ ℝ d define the function τ r : ℤ d → ℤ d , z = ( z 0 , ... , z d - 1 ) ↦ ( z 1 , ... , z d - 1 , - rz ) , where rz is the scalar product of the vectors r and z . If each orbit of τ r ends up at 0 , we call τ r a shift radix system. It is a well-known fact that each orbit of τ r ends up periodically if the polynomial t d + r d - 1 t d - 1 + ⋯ + r 0 associated to r is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of...

Fonctions digitales le long des nombres premiers

Bruno Martin, Christian Mauduit, Joël Rivat (2015)

Acta Arithmetica

In a recent work we gave some estimations for exponential sums of the form ∑ n ≤ x Λ ( n ) e x p ( 2 i π ( f ( n ) + β n ) ) , where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.

Generalized golden ratios of ternary alphabets

Vilmos Komornik, Anna Chiara Lai, Marco Pedicini (2011)

Journal of the European Mathematical Society

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence...

Greedy and lazy representations in negative base systems

Tomáš Hejda, Zuzana Masáková, Edita Pelantová (2013)

Kybernetika

We consider positional numeration systems with negative real base - β , where β > 1 , and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal ( - β ) -representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base β 2 with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...

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