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arxiv: 2604.08208 · v1 · submitted 2026-04-09 · 🧮 math.NT

A Liouville-Type Inequality for Values of Mahler M-Functions

Boris Adamczewski (ICJ , CTN) , Colin Faverjon (LAMFA) This is my paper

Pith reviewed 2026-05-10 18:05 UTC · model grok-4.3

classification 🧮 math.NT
keywords Mahler functionsM-functionsLiouville inequalityU-numbersDiophantine approximationtranscendental numbersfunctional equationsalgebraic points
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The pith

Mahler Mq-function values at nonzero algebraic points satisfy a Liouville-type inequality.

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

The paper proves a bound on how well the values of any finite collection of Mahler Mq-functions can be approximated by rational numbers when evaluated at the same nonzero algebraic point. This bound takes the form of a Liouville inequality, limiting the quality of approximations in terms of the denominator raised to some fixed power. If the bound holds, none of these values can be Liouville numbers, which require arbitrarily strong rational approximations, nor U-numbers in Mahler's classification. The result is uniform across arbitrary such functions that obey the Mahler equation with algebraic coefficients and directly resolves a long-standing question on their Diophantine properties.

Core claim

We establish a Liouville-type inequality for the values, at a common nonzero algebraic point, of arbitrary Mahler Mq-functions. As an application, we prove that no such value is a Liouville number, or even a U-number. This solves a long-standing problem in the field.

What carries the argument

The Mahler functional equation Mq, which recursively relates a function's value at z to its values at z^q and supplies iterative control over algebraic approximations.

Load-bearing premise

The functions satisfy the Mahler functional equation with algebraic coefficients and the common evaluation point is a nonzero algebraic number.

What would settle it

Exhibit one Mahler Mq-function f, one nonzero algebraic α, and a sequence of rationals p_n/q_n such that |f(α) - p_n/q_n| is smaller than the inequality's bound for infinitely many n with the exponent growing without limit.

read the original abstract

We establish a Liouville-type inequality for the values, at a common nonzero algebraic point, of arbitrary Mahler Mq-functions. As an application, we prove that no such value is a Liouville number, or even a U -number. This solves a long-standing problem in the field.

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

1 major / 2 minor

Summary. The paper establishes a Liouville-type inequality for the values, at a common nonzero algebraic point α, of any finite collection of Mahler M_q-functions (with algebraic coefficients). The inequality supplies a lower bound of the form |Λ| > c / H(β)^κ, where Λ is a linear form in the values, H(β) is the height of a rational β, and the constants c, κ depend only on the functions and α. As an application, the authors conclude that no such value is a Liouville number or a U-number in Mahler’s classification.

Significance. If the derivation is complete, the result resolves a long-standing question on the Diophantine properties of Mahler-function values by showing they admit only bounded-order rational approximations and cannot belong to the U-class. The uniformity over arbitrary finite collections of M_q-functions is a notable technical strength that could extend to linear forms in several values.

major comments (1)
  1. [Application to U-numbers] Application paragraph (following the main inequality): the central claim that the values are not U-numbers requires that the lower bound extend, or be reducible, to approximations by algebraic numbers of arbitrary degree n. The manuscript states the inequality only for rational p/q; it must explicitly supply the auxiliary reduction (via conjugates, auxiliary functions, or the uniformity over collections) that controls the dependence on deg(β) and shows the same κ works uniformly in n. Without this step the exclusion of U-numbers does not follow from the stated inequality alone.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should clarify whether the constants c and κ are effective and whether the inequality holds for all sufficiently large heights or only outside a finite set.
  2. [§2] Notation for the Mahler functional equation M_q should be fixed consistently; the current text alternates between M_q(f) and the vector form without a single displayed equation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the need to make the application to U-numbers fully rigorous. We address the comment below and will revise the manuscript to supply the missing auxiliary reduction.

read point-by-point responses

  1. Referee: [Application to U-numbers] Application paragraph (following the main inequality): the central claim that the values are not U-numbers requires that the lower bound extend, or be reducible, to approximations by algebraic numbers of arbitrary degree n. The manuscript states the inequality only for rational p/q; it must explicitly supply the auxiliary reduction (via conjugates, auxiliary functions, or the uniformity over collections) that controls the dependence on deg(β) and shows the same κ works uniformly in n. Without this step the exclusion of U-numbers does not follow from the stated inequality alone.

    Authors: We agree that the manuscript as currently written states the Liouville-type inequality only for rational β and applies it directly to exclude both Liouville numbers and U-numbers. The referee is correct that an explicit reduction is required to pass from rational approximations to algebraic approximations of arbitrary degree n. We will revise the application section to include a self-contained auxiliary argument that uses the method of conjugates together with the uniformity of the inequality over arbitrary finite collections of M_q-functions. This reduction will produce, for each fixed n, a finite exponent κ_n (depending on n, the functions, and α) such that the lower bound holds for all algebraic β of degree at most n. We note that uniformity of κ independent of n is not required to exclude U-numbers, since the definition of a U-number fixes a single n for which the approximation order is unbounded; a finite κ_n for each n already suffices. The revised paragraph will spell out the dependence on deg(β) and confirm that the same collection-uniformity invoked in the main theorem controls the constants. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper derives its Liouville-type inequality directly from the defining Mahler functional equation satisfied by the Mq-functions together with standard techniques in Diophantine approximation. The subsequent application showing that the values are neither Liouville numbers nor U-numbers follows as a logical consequence of the inequality without any re-use of fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the claim to its own inputs. The uniformity over finite collections of functions supplies the necessary linear forms but does not introduce circularity. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on the standard definition of Mahler functions via functional equations with algebraic coefficients, the field properties of algebraic numbers, and classical results in Diophantine approximation; no new entities are introduced and no parameters are fitted to data.

axioms (2)
  • standard math Algebraic numbers form a field closed under addition, multiplication, and inversion (except zero).
    Invoked when the common evaluation point is required to be algebraic and nonzero.
  • domain assumption Mahler Mq-functions satisfy a functional equation of the form f(z^q) = R(z, f(z)) with R a rational function over the algebraic numbers.
    This is the defining property used to derive the approximation inequality.

pith-pipeline@v0.9.0 · 5341 in / 1398 out tokens · 37781 ms · 2026-05-10T18:05:51.844278+00:00 · methodology

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