Multiple expansions of real numbers with digits set $$\left\{ 0,1,q\right\} $$ 0 , 1 , q

Β-expansions with Deleted Digits for Pisot Numbers Β

An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ(β,D) of real numbers with representations x = ∑∞ n=1 dnβ −n, where each dn ∈ D, a finite set of “digits”, and β > 0 is a Pisot number. The Hausdorff dimension is shown to be log λ/ log β, where λ is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data β,D is given.

beta-EXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta

Expansions of Real Numbers

It was discovered some years ago that there exist non-integer real numbers q > 1 for which only one sequence (ci) of integers 0 ≤ ci < q satisfies the equality P ∞ i=1 ciq −i = 1. The set U of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. For each fi...

Unique Expansions of Real Numbers

It was discovered some years ago that there exist non-integer real numbers q > 1 for which only one sequence (ci) of integers 0 ≤ ci < q satisfies the equality P∞ i=1 ciq −i = 1. The set U of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. For each fix...

Random Β-expansions with Deleted Digits

In this paper we define random β-expansions with digits taken from a given set of real numbers A = {a1, . . . , am}. We study a generalization of the greedy and lazy expansion and define a function K, that generates essentially all β-expansions with digits belonging to the set A. We show that K admits an invariant measure ν under which K is isomorphic to the uniform Bernoulli shift on A.