Chains of Cantor subspaces of Mahler T-Numbers and the middle-third Cantor set

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Abstract

We investigate Mahler’s real T-numbers through Cantor-type constructions inside $\mathcal{T} \cap \mathbb{G}$, where G is the middle-third Cantor set and $\mathcal{T}$ is the set of real Mahler T-numbers. We build explicit families $\mathcal{T}$^(t) which are homeomorphic to Cantor space and use them to analyze structural, combinatorial, and additive properties of $\mathcal{T}$ . Our results include the existence of descending chains of Cantor subsets of $\mathcal{T}$ of length c; a characterization of ternary expansions in $\mathcal{T}$^(t), showing non- normality but maximal block complexity under sparse forcing, linked to the Adamczewski–Bugeaud criterion; and a sumset theorem proving that for suitable parameters t₁, t₂ one has the interval identity $\mathcal{T}$^(t1) + $\mathcal{T}$^(t2) = [0, 1], which yields the global corollary $\mathcal{T}$ + $\mathcal{T}$ = $\mathbb{R}$ (Erd˝os property) by integer translation invariance. We also discuss implications for cardinal invariants and entropy of the shift map, highlighting the interplay between thin Diophantine sets and large additive structure. To address a natural concern about existence, we include a non-emptiness lemma which shows that our scheduled deletion-and-witness procedure always leaves a non-empty perfect set $\mathcal{T}$ (t).

Keywords

Mahler T-numbers / Cantor set / descending chain / cardinality / Erd˝os property

CHANGE HISTORY

A Correction to this paper has been published:
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The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
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References

[1] K. Mahler, Zur Approximation algebraischer Zahlen. I, Math. Ann., 107 (1933), 691-730. https://doi.org/10.1007/BF01448915

[2] T. P. Chalebgwa, S. A. Morris, Erd˝os-Liouville Sets, Bull. Aust. Math. Soc., 107 (2023), 284-289. https://doi.org/10.1017/S0004972722001009

[3] A. Baker, On Mahler’s classification of transcendental numbers, Acta Math., 111 (1964), 97-120. https://doi.org/10.1007/BF02391010

[4] B. Adamczewski, Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in integer bases, Ann. Math., 165 (2007), 547-565. https://doi.org/10.4007/annals.2007.165.547