Unique expansions of real numbers

Vries, Martijn de; Komornik, Vilmos

doi:10.1016/j.aim.2008.12.008

Cited by 130 publications

(186 citation statements)

References 27 publications

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“…On the other hand, the set of numbers x having a unique expansion has many unexpected topological and combinatorial properties, depending on the value of q; see, e.g., Daróczy and Kátai [1], de Vries [2], [3], Glendinning and Sidorov [7], and [4]. Given a finite alphabet A = {a 1 , .…”

Section: Introductionmentioning

confidence: 99%

Generalized golden ratios of ternary alphabets

Komornik¹,

Lai²,

Pedicini³

2011

J. Eur. Math. Soc.

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Abstract. 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 on the alphabets.

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Section: Introductionmentioning

confidence: 99%

Generalized golden ratios of ternary alphabets

Komornik¹,

Lai²,

Pedicini³

2011

J. Eur. Math. Soc.

Self Cite

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show abstract

“…Finally, we recall some topological properties of U and U which were essentially established in [8,18] (see also, [9]). …”

Section: Lemma 24 Let Q ∈ (1 M + 1) Then Q ∈ U If and Only Ifmentioning

confidence: 99%

“…, M}. For instance, it was shown in [8] that U q is closed if and only if q does not belong to the set U. It is well-known that U is a Lebesgue null set of full Hausdorff dimension (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Many works are devoted to the univoque sets U q (see, e.g., [10,11,14]). Recently, de Vries and Komornik investigated their topological properties in [8]. Komornik et al considered their Hausdorff dimension in [19], and showed that the dimension function D : q → dim H U q behaves like a Devil's staircase on (1, M +1].…”

Section: Introductionmentioning

confidence: 99%

“…[8]). Furthermore, each left endpoint q 0 is algebraic, while each right endpoint q * 0 ∈ U is transcendental (cf.…”

Section: Introductionmentioning

confidence: 99%

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