Abstract
For q>1 we consider expansions in base q with digits set left{ 0,1,qright} . Let {{mathcal {U}}}_q be the set of points which have a unique q-expansion. For k=2, 3,ldots ,aleph _0 let mathcal {B}_k be the set of bases q>1 for which there exists x having precisely k different q-expansions, and for qin mathcal {B}_k let {{mathcal {U}}}_q^{(k)} be the set of all such x’s which have exactly k different q-expansions. In this paper we show that Bℵ0=[2,∞)andBk=(qc,∞)for anyk≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathcal {B}_{\\aleph _0}=[2,\\infty )\\quad \\text {and}\\quad \\mathcal {B}_k=(q_c,\\infty )\\quad \\text {for any}\\quad k\\ge 2, \\end{aligned}$$\\end{document}where q_capprox 2.32472 is the appropriate root of x^3-3x^2+2x-1=0. Moreover, we show that for any integer kge 2 and any qin mathcal {B}_{k} the Hausdorff dimensions of {{mathcal {U}}}_q^{(k)} and {{mathcal {U}}}_q are the same, i.e., dimHUq(k)=dimHUqfor anyk≥2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\dim _H{{\\mathcal {U}}}_q^{(k)}=\\dim _H{{\\mathcal {U}}}_q\\quad \\text {for any}\\quad k\\ge 2. \\end{aligned}$$\\end{document}Finally, we conclude that the set of points having a continuum of q-expansions has full Hausdorff dimension.
Highlights
Expansions in non-integer bases were pioneered by Rényi [18] and Parry [16]
For a given β ∈ (1, 2), it is well-known that typically a real number x ∈
For q > 1 and k ∈ N we recall that Uq(k) is the set of x ∈ [0, q/(q − 1)] having precisely k different q-expansions
Summary
Expansions in non-integer bases were pioneered by Rényi [18] and Parry [16]. Unlike integer base expansions, for a given β ∈ (1, 2), it is well-known that typically a real number x ∈. Zou et al [24] considered the set of points in Eq which have a unique q-expansion. We investigate the set of points in Eq having multiple q-expansions. (ii) For any integer k ≥ 2 and any q ∈ Bk we have dimH Uq(k) = dimH Uq. As a result of Theorem 2 it follows that qc is the critical base, in the sense that Uq(k) has positive Hausdorff dimension if q > qc, while Uq(k) has zero Hausdorff dimension if q ≤ qc. (ii) For any q > 1 we have dimH Uq(2א0 ) = dimH Eq. Remark 1.2 In Lemma 5.5 we prove a stronger result of Theorem 3 (ii), and show that the Hausdorff measures of Uq(2א0 ) and Eq are the same for any q > 1, i.e., Hs Uq2א0 = Hs (Eq ) ∈ (0, ∞), where s = dimH Eq. The rest of the paper is arranged as follows.
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