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OP-9654Status unassessedNumber theory

Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers

Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers

MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

problem_10_9

Problem 10.9. There are no real numbers ξ\xi such that 0≤{ξ(3/2)n}<1/20 \le \{\xi (3/2)^n\} < 1/2 for every positive integer nn, i.e. no Z-number exists. Posed by Mahler [Mah68].

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Formal statement

Lean 4
/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil
import FormalConjectures.Wikipedia.Mahler32
/-!
# Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers

See also FormalConjectures/Wikipedia/Mahler32.lean.

*References:*
  - [Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation."
    Vol. 193. Cambridge University Press, 2012. Chapter 10.
  - [Mah68] Mahler, Kurt. "An unsolved problem on the powers of 3/2."
    Journal of the Australian Mathematical Society 8.2 (1968): 313-321.
  - [FLP95] Flatto, Leopold, Jeffrey C. Lagarias, and Andrew D. Pollington.
    "On the range of fractional parts $\{\xi(p/q)^n\}$."
    Acta Arithmetica 70.2 (1995): 125-147.
-/

namespace Bugeaud09

/--
Problem 10.9. There are no real numbers $\xi$ such that $0 \le \{\xi (3/2)^n\} < 1/2$
for every positive integer $n$, i.e. no Z-number exists. Posed by Mahler [Mah68].
-/
@[category research open, AMS 11]
theorem problem_10_9 : type_of% Mahler32.mahler_conjecture := by
  sorry

/--
Sanity check: `1` is not a Z-number, since for `n = 1` we have
$\{1 \cdot (3/2)^1\} = \{3/2\} = 1/2 \not< 1/2$.
-/
@[category test, AMS 11]
theorem not_isZNumber_one : ¬ IsZNumber 1 := by
  rintro ⟨-, h⟩
  have h1 := h 1 le_rfl
  rw [pow_one, one_mul] at h1
  have hfr : Int.fract (3 / 2 : ℝ) = 1 / 2 := by rw [Int.fract]; norm_num [Int.floor_eq_iff]
  rw [hfr] at h1
  exact lt_irrefl _ h1

/--
Mahler [Mah68] proved that the set of Z-numbers has Lebesgue measure zero.
-/
@[category research solved, AMS 11]
theorem problem_10_9.variants.measure_zero :
    MeasureTheory.volume {ξ : ℝ | IsZNumber ξ} = 0 := by
  sorry

/--
Flatto, Lagarias, and Pollington [FLP95] proved that the set of Z-numbers has
Hausdorff dimension strictly less than $1$.
-/
@[category research solved, AMS 11]
theorem problem_10_9.variants.dimH_lt_one :
    dimH {ξ : ℝ | IsZNumber ξ} < 1 := by
  sorry

end Bugeaud09

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.

Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.

References
  1. [1]Formal Conjectures — Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers link

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