A property of algebraic univoque numbers

Vries, Martijn de

doi:10.1007/s10474-007-6252-x

Cited by 7 publications

(4 citation statements)

References 11 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We already know from the above proof that the second inequality is always strict. Assume on the contrary that 0d 2 We have obtained in this way a new characterization of Sturmian sequences: a sequence s is Sturmian if and only if 1s is an admissible sequence defined by an infinite sequence h = (h i ) such that h i ≥ 2 for infinitely many i. …”

Section: Examples 32mentioning

confidence: 99%

See 1 more Smart Citation

Generalized golden ratios of ternary alphabets

Komornik¹,

Lai²,

Pedicini³

2011

J. Eur. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Section: Examples 32mentioning

confidence: 99%

“…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.

View full text Add to dashboard Cite

show abstract

“…We refer to the papers [23], [6], [7], [8], [9], [3] and surveys [32], [20] and [10] for more information.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of univoque sets and Devil's staircase

Komornik

Kong

2017

Advances in Mathematics

111

View full text Add to dashboard Cite

Abstract. We fix a positive integer M , and we consider expansions in arbitrary real bases q > 1 over the alphabet {0, 1, . . . , M }. We denote by U q the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of U q for each q ∈ (1, ∞). Furthermore, we prove that the dimension function D : (1, ∞) → [0, 1] is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in (q ′ , ∞), where. During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in β-expansions, and we determine for all M the Lebesgue measure and the Hausdorff dimension of the set U of bases in which x = 1 has a unique expansion.

show abstract

“…Then b p (x) := (b i ) is an expansion of x in base p. 2 By construction this is the lexicographically largest expansion of x in base p, called the greedy or β-expansion of x in base p. Today there is a huge literature devoted to non-integer expansions. For example, probabilistic and ergodic aspects are investigated in [3], [5], [31], [32], [34], combinatorial properties in [1], [4], [18], [25], [26], [30], unique expansions in [6], [7], [8], [9], [10], [12], [14], [15], [16], [21], [22], [23], [24], [27], and control-theoretical applications are given in [2], [28], [29].…”

Section: Introductionmentioning

confidence: 99%