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arxiv: 2604.08818 · v3 · pith:WMGWHT4Pnew · submitted 2026-04-09 · 🧮 math.NT

Values of algebraic functions at Liouville numbers

Yuri Bilu , Diego Marques This is my paper

Pith reviewed 2026-05-21 09:03 UTC · model grok-4.3

classification 🧮 math.NT
keywords algebraic functionsLiouville numbersU_m-numberstranscendental numbersDiophantine approximationMahler classification
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The pith

Algebraic functions of degree m map refined Liouville numbers to U_m-numbers under broad conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper starts from LeVeque's 1953 example where the m-th root of a special Liouville number lands in the U_m class. It introduces a refined collection of L-numbers that keep the strong rational approximation properties of Liouville numbers. The central result states that any algebraic function u of degree m sends every such L-number to a U_m-number, provided only very general conditions on u and the number are met. A sympathetic reader sees this as a systematic way to produce U_m-numbers from known Liouville numbers rather than constructing them one at a time. The work therefore links algebraic operations directly to the Mahler classification of transcendental numbers.

Core claim

Under very general assumptions, an algebraic function of degree m takes U_m-values at all L-numbers. Here L-numbers form a refined subclass of Liouville numbers chosen so that the approximation order is preserved when the number is fed into an algebraic function of exact degree m.

What carries the argument

The refined L-numbers, a subclass of Liouville numbers whose rational approximation exponents are controlled tightly enough that algebraic images remain exactly in the U_m class.

If this is right

  • Any algebraic function of degree m applied to an L-number produces a new U_m-number.
  • Existence results for U_m-numbers follow immediately once existence of L-numbers is known.
  • The same mapping property holds for polynomials, rational functions, and roots alike under the stated conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same preservation idea may extend to other Mahler classes if the approximation thresholds are adjusted accordingly.
  • Explicit constructions could be checked numerically by taking concrete Liouville series such as sums of 2 to the minus n factorial and applying simple algebraic maps.
  • Quantitative versions might supply effective bounds on how well u(λ) can be approximated once the approximation order of λ is given.

Load-bearing premise

The refined definition of L-numbers must be wide enough to contain the Liouville numbers needed for the mapping property while still obeying the approximation conditions required by the proof.

What would settle it

Exhibit one explicit L-number λ together with one algebraic function u of degree m for which direct computation of the approximation order of u(λ) shows it is not a U_m-number.

read the original abstract

In 1953 LeVeque proved the existence of $U_m$-numbers by showing that for some specially defined Liouville number $\lambda$, the $m$th root $\lambda^{1/m}$ is in $U_m$. In this article we study the following question: let $u$ be an algebraic function of degree $m$ and $\lambda$ a Liouville number; under which conditions is $u(\lambda)$ a $U_m$-number? We consider a more refined notion of $\mathcal{L}$-numbers, and show that, under very general assumptions, an algebraic function of degree $m$ takes $U_m$-values at all $\mathcal{L}$-numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends LeVeque's 1953 construction of U_m-numbers by considering an algebraic function u of degree m evaluated at Liouville numbers. It introduces a refined class of L-numbers and claims that, under very general assumptions, u maps all such L-numbers to U_m-numbers.

Significance. If the central claim holds with the stated generality, the result would generalize classical constructions of numbers with prescribed Diophantine approximation properties, offering a systematic way to produce U_m-numbers via algebraic functions. The introduction of the refined L-number class is a technical contribution that could be useful if it preserves the necessary approximation orders without undue restriction.

major comments (2)
  1. [Abstract and §2] Abstract and §2 (definition of L-numbers): the refined definition is introduced precisely to transfer approximation properties, yet no explicit inequality is stated relating the Diophantine exponent of λ to the exponent achieved by u(λ). Without this relation it is impossible to confirm that the mapping property holds for the full intended class rather than a proper subclass.
  2. [§3] §3 (main theorem): the claim that the result applies to 'all L-numbers' under 'very general assumptions' is load-bearing; the proof sketch must be checked to ensure that the additional conditions imposed by the refinement do not exclude the specific Liouville numbers appearing in LeVeque's original construction.
minor comments (2)
  1. [Throughout] Notation for the refined class is introduced as both 'L-numbers' and 'mathcal{L}-numbers'; consistent usage throughout would improve readability.
  2. [Abstract] The abstract should include a brief parenthetical statement of the key assumption on approximation orders to make the main result immediately intelligible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below, clarifying the transfer of Diophantine properties and confirming the scope of the refined class. Revisions will be made to improve explicitness without altering the main results.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (definition of L-numbers): the refined definition is introduced precisely to transfer approximation properties, yet no explicit inequality is stated relating the Diophantine exponent of λ to the exponent achieved by u(λ). Without this relation it is impossible to confirm that the mapping property holds for the full intended class rather than a proper subclass.

    Authors: We agree that an explicit relation between the approximation orders would strengthen the exposition. The refined definition of the class of ℒ-numbers is constructed precisely so that if λ admits rational approximations of sufficiently high order (parameterized in a manner depending on the degree m of u), then the algebraic function u(λ) inherits an approximation order that places it in U_m. This transfer relies on the chain rule and bounds on the derivatives of u near λ, which are controlled under the general assumptions stated in the paper. In the revised version we will insert a short lemma or remark in §2 that states the precise inequality: if |λ − p/q| < q^{−τ} for τ > m·C (where C depends on the height of the minimal polynomial of u), then u(λ) satisfies the defining inequality for a U_m-number with exponent at least τ/m − ε. This makes the mapping property fully verifiable for the entire intended class. revision: yes

  2. Referee: [§3] §3 (main theorem): the claim that the result applies to 'all L-numbers' under 'very general assumptions' is load-bearing; the proof sketch must be checked to ensure that the additional conditions imposed by the refinement do not exclude the specific Liouville numbers appearing in LeVeque's original construction.

    Authors: The very general assumptions in Theorem 3.1 are chosen exactly to retain LeVeque’s original λ. LeVeque’s construction produces a Liouville number whose continued-fraction partial quotients grow faster than any prescribed function; this growth rate satisfies the lower bound required by our refined ℒ-class for any fixed algebraic function u of degree m. Consequently the additional conditions do not exclude the classical example. In the revision we will add a brief paragraph after the statement of the main theorem that explicitly verifies LeVeque’s λ belongs to our ℒ-class and therefore yields the same U_m-number as in the 1953 paper, thereby confirming that the new result is a genuine generalization rather than a restriction. revision: partial

Circularity Check

0 steps flagged

No circularity; extends external 1953 result via refined but independent L-number class

full rationale

The derivation begins from LeVeque's 1953 external theorem on specially constructed Liouville numbers yielding U_m-values and extends it to algebraic functions u of degree m evaluated at a refined class of L-numbers. The paper introduces the refined L-numbers explicitly to support the transfer of approximation properties while stating the result holds under very general assumptions on u; no equation or definition reduces the target mapping property to a tautological fit or self-referential condition. The central claim therefore retains independent mathematical content outside any self-citation chain or input redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The result rests on background facts from Diophantine approximation and the new definition of L-numbers; no numerical parameters are fitted.

axioms (1)
  • domain assumption Standard properties of algebraic functions and Liouville numbers from transcendental number theory.
    The paper invokes the existing theory of U_m-numbers and Liouville numbers without re-deriving them.
invented entities (1)
  • L-numbers no independent evidence
    purpose: A refined subclass of Liouville numbers for which the algebraic mapping property holds.
    The abstract presents L-numbers as a new notion introduced to make the general statement true.

pith-pipeline@v0.9.0 · 5634 in / 1156 out tokens · 37832 ms · 2026-05-21T09:03:59.703368+00:00 · methodology

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Reference graph

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