Mahler Problem: Arithmetic Rigidity and Beyond

• Mahler Problem is a collection of challenges in number theory involving functional equations, arithmetic rigidity, and self-similar structures.
• The topic encompasses entire transcendental functions with controlled arithmetic images, dual-base rigidity in Mahler functions, and algorithmic methods for solving related equations.
• Applications range from optimizing Mahler measure in polynomial settings to addressing convex-geometric conjectures and intrinsic approximation on Cantor sets.

In current arXiv-facing usage, the expression Mahler Problem is not a single theorem but a family of problems initiated by Kurt Mahler and later extended across transcendence theory, Mahler functional equations, Mahler measure, convex geometry, and Diophantine approximation. The common theme is arithmetic rigidity under structures that Mahler introduced or emphasized: functional equations of the form $f(z)\mapsto f(z^k)$, arithmetic images of exceptional sets such as $\overline{\mathbb{Q}}$ or the Liouville numbers, the Mahler measure of polynomials, the Mahler volume product of convex bodies, and approximation phenomena on self-similar sets. Several of these problems have now been solved in strong forms, while others remain conjectural or only partially understood (Alves, 2023, Adamczewski et al., 2013, Mossinghoff, 2024, Harrell et al., 2011, Broderick et al., 2011).

Domain	Representative formulation	Status in the cited literature
Entire functions and algebraic values	Find transcendental entire $f$ with $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$ and $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$	Solved; derivative variants also constructed (Alves, 2023)
Mahler functions	If $F$ is both $k$- and $\ell$-Mahler with $k,\ell$ multiplicatively independent, must $F$ be rational?	Yes in characteristic $\overline{\mathbb{Q}}$0 (Adamczewski et al., 2013)
Mahler measure of $\overline{\mathbb{Q}}$1 polynomials	Maximize $\overline{\mathbb{Q}}$2 for high-degree Littlewood polynomials	New asymptotic record $\overline{\mathbb{Q}}$3 via Turyn polynomials (Mossinghoff, 2024)
Mahler volume	Minimize $\overline{\mathbb{Q}}$4 for symmetric convex bodies	Local structure clarified; in dimension $\overline{\mathbb{Q}}$5, local minimizers are parallelograms (Harrell et al., 2011)
Powers near integers	Classify algebraic $\overline{\mathbb{Q}}$6 with infinitely many $\overline{\mathbb{Q}}$7	Equivalent to $\overline{\mathbb{Q}}$8 being a PV number for some $\overline{\mathbb{Q}}$9 (Philippon et al., 2019)
Cantor-set approximation	Approximate irrational $f$0 by rationals in $f$1	Dirichlet-type theorem proved (Broderick et al., 2011)

1. Entire transcendental functions and arithmetic images

A classical Mahler question asks for transcendental entire functions with controlled arithmetic behavior on algebraic points. In the set-theoretic interpretation used in recent work, Mahler’s Problem B asks for a transcendental entire function with rational coefficients such that

$f$2

where $f$3 is the set-theoretic preimage. This interpretation is essential because an entire transcendental function cannot admit a global functional inverse, by Picard’s theorem. The same source places this question in the lineage of Stäckel’s local 1902 construction and records that Marques–Moreira (2017) obtained the stronger equalities

$f$4

again for transcendental entire functions with rational coefficients (Alves, 2023).

The derivative-enriched variant asks whether one can impose the same arithmetic image and preimage conditions simultaneously on all derivatives. The answer is affirmative in a strong existence form: there are uncountably many transcendental entire functions $f$5 with rational coefficients such that for every $f$6,

$f$7

The construction is inductive rather than closed-form. It enumerates algebraic numbers in a conjugation-stable pattern, starts from $f$8, and adds perturbations $f$9 with carefully chosen real algebraic $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$0 and algebraic-coefficient polynomials $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$1. Rouché’s theorem and Hurwitz’s theorem control zeros and preimages under perturbation, while the Weierstrass M-test yields uniform convergence on compacta and hence entireness together with convergence of all derivatives. A separate bookkeeping argument ensures that each Taylor coefficient is eventually fixed as a rational number (Alves, 2023).

A different problem posed by Mahler concerns Liouville numbers. Maillet had proved that nonconstant rational functions with rational coefficients preserve the set of Liouville numbers. Mahler asked whether nonconstant entire transcendental functions can exhibit analogous behavior. For large parametrized subclasses $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$2 of the Liouville numbers, there are uncountably many entire transcendental functions

$$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$3

such that for every $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$4 one has $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$5, $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$6, and $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$7. The construction uses rational Taylor coefficients with divisibility chains, transport estimates for rational approximants, and a coprimeness device that excludes rational outputs. The result is formulated for large parametrized subclasses rather than for all Liouville numbers (Marques et al., 2015).

2. Mahler functions, dual-base rigidity, and automatic structures

A $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$8-Mahler function is a power series $$f(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$9 such that the family

$$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$0

is linearly dependent over $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$1. Equivalently, $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$2 satisfies a Mahler-type functional equation

$$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$3

with coefficients in $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$4. A central rigidity theorem of Adamczewski and Bell resolves a conjecture of Loxton and van der Poorten: if $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$5 has characteristic $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$6 and $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$7 are multiplicatively independent, then a power series $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$8 is both $$f^{-1}(\overline{\mathbb{Q}})\subseteq\overline{\mathbb{Q}}$$9-Mahler and $F$0-Mahler if and only if $F$1 is rational (Adamczewski et al., 2013). The proof combines structural normalization of Mahler equations, decomposition into regular parts and infinite products, elimination of cyclotomic singularities, reduction modulo infinitely many primes via Chebotarev’s density theorem, and Cobham’s theorem on simultaneous automaticity.

This one-variable dual-base theorem sits inside a broader multivariate theory. Linear Mahler systems in several variables are written

$F$2

with $F$3 and $F$4 an integer matrix with nonnegative entries. A principal result is a lifting theorem: under admissibility of the pair $F$5 and regularity of the algebraic point $F$6, any homogeneous algebraic relation among the values $F$7 lifts to a homogeneous algebraic relation over $F$8 among $F$9. As a corollary,

$k$0

Purity theorems then separate algebraic relations coming from independent points and from multiplicatively independent spectral radii, yielding strong algebraic independence statements for values of Mahler functions and, in particular, new proofs and generalizations of Cobham-type results for automatic numbers (Adamczewski et al., 2020, Adamczewski et al., 2018).

The differential-algebraic side of the subject studies whether Mahler functions and their derivatives are algebraically independent. For systems

$k$1

parametrized difference Galois theory converts differential-algebraic relations among solutions into structure theorems for linear differential algebraic groups. When the classical difference Galois group contains $k$2 and $k$3 is a monomial, any nonzero solution column is hyperalgebraically independent over $k$4. This yields, for example, hyperalgebraic independence of the Baum–Sweet and Rudin–Shapiro generating functions together with all of their derivatives (Dreyfus et al., 2015).

The computational side is now developed as well. For scalar equations

$k$5

and systems

$k$6

algorithms are available for power-series, polynomial, rational, Puiseux, Hahn, and full mixed solutions. One line of work computes rational and polynomial solutions and a gcrd in the Ore algebra of Mahler operators, with polynomial-time complexity under a mild assumption (Chyzak et al., 2016). A later algorithmic advance computes a complete basis of solutions for any Mahler equation together with a fundamental matrix of any Mahler system, in a structured form

$k$7

where $k$8 is the Puiseux part, $k$9 is built from explicit Hahn series, and $\ell$0 encodes constant-exponential solutions of $\ell$1 (Faverjon et al., 24 Nov 2025). Parallel to this, Mahler discrete residues and their twisted analogues solve the summability problems

$\ell$2

for rational functions by giving complete obstruction theories in terms of orbitwise residue data on the multiplicative dynamics $\ell$3 (Arreche et al., 2022, Arreche et al., 2023).

3. Mahler measure: limits, optimization, dynamics, and $\ell$4-values

For a nonzero Laurent polynomial $\ell$5, the logarithmic Mahler measure is

$\ell$6

with the one-variable specialization

$\ell$7

One important problem studies stability of Mahler measure under monomial substitutions. If $\ell$8 is a sequence of integer matrices with $\ell$9, where $k,\ell$0 is the minimum $k,\ell$1-norm of a nonzero vector in $k,\ell$2, then for every nonzero Laurent polynomial $k,\ell$3,

$k,\ell$4

The same work proves an explicit error term

$k,\ell$5

for $k,\ell$6, and in the torus-zero-free case the convergence is exponentially fast (Brunault et al., 2022).

Another line of work uses Mahler measure as an extremal functional on Littlewood polynomials. Here Mahler’s problem asks for the largest possible value of the normalized ratio

$k,\ell$7

for polynomials with $k,\ell$8 coefficients and large degree. Since $k,\ell$9, one always has $F$0. For Turyn polynomials

$F$1

obtained by cyclically shifting the Legendre symbol sequence, the asymptotic normalized Mahler measure is described by a random process $F$2 and its expectation functional

$F$3

The computations indicate a maximum near the quarter shift $F$4, and the resulting Littlewood companions satisfy

$F$5

establishing a new record asymptotic value $F$6 for high-degree $F$7 polynomials (Mossinghoff, 2024).

Mahler measure also defines a dynamical map on algebraic numbers. If $F$8 has minimal polynomial $F$9, its multiplicative Mahler measure is

$\overline{\mathbb{Q}}$00

Iterating $\overline{\mathbb{Q}}$01 leads to periodic, preperiodic, and wandering points. There are no points of strict period greater than $\overline{\mathbb{Q}}$02; fixed points are natural numbers, Pisot numbers, or Salem numbers. For abelian number fields, all elements are preperiodic if and only if the maximal totally real subfield has Galois group $\overline{\mathbb{Q}}$03, $\overline{\mathbb{Q}}$04, $\overline{\mathbb{Q}}$05, or $\overline{\mathbb{Q}}$06, and every degree-$\overline{\mathbb{Q}}$07 extension of $\overline{\mathbb{Q}}$08 contains a wandering algebraic unit (Fili et al., 2021).

The measure also interacts with special values of $\overline{\mathbb{Q}}$09-functions. For

$\overline{\mathbb{Q}}$10

the corresponding elliptic curve is

$\overline{\mathbb{Q}}$11

Using a Kronecker–Eisenstein series formula of Rodriguez Villegas together with CM-point analysis, one obtains 28 new identities of the form

$\overline{\mathbb{Q}}$12

with $\overline{\mathbb{Q}}$13 a weight-$\overline{\mathbb{Q}}$14 cusp form. A further regulator calculation proves five determinant formulas expressing $\overline{\mathbb{Q}}$15 for CM elliptic curves over real quadratic fields as $\overline{\mathbb{Q}}$16 determinants of Mahler measures, extending earlier results for $\overline{\mathbb{Q}}$17 (Tao et al., 2022).

A more geometric extension links Mahler measure to amoebae of Laurent polynomials. For certain Newton polynomials arising in dimer and quiver settings, the volume of the bounded complement of a $\overline{\mathbb{Q}}$18-dimensional amoeba is empirically related to the gas-phase contribution to Mahler measure by an approximately degree-$\overline{\mathbb{Q}}$19 polynomial, with $\overline{\mathbb{Q}}$20 and $\overline{\mathbb{Q}}$21. The Ronkin function supplies the bridge, since $\overline{\mathbb{Q}}$22 and the amoeba boundary marks the transition from linear to strictly convex behavior of $\overline{\mathbb{Q}}$23 (Chen et al., 2022).

4. Mahler volume and the convex-geometric conjecture

In convex geometry, the Mahler volume or volume product of a convex body $\overline{\mathbb{Q}}$24 containing the origin is

$\overline{\mathbb{Q}}$25

For origin-symmetric bodies, Mahler conjectured that the cube minimizes $\overline{\mathbb{Q}}$26; equivalently, among symmetric convex bodies,

$\overline{\mathbb{Q}}$27

should be minimal, with equality only for affine images of the cube and its polar, up to duality. The Blaschke–Santaló inequality gives the opposite extremum, asserting that ellipsoids maximize $\overline{\mathbb{Q}}$28 (Harrell et al., 2011).

A local analysis of the support-function formulation of $\overline{\mathbb{Q}}$29 yields first- and second-variation formulas. On a $\overline{\mathbb{Q}}$30 patch with positive Gauss curvature, the second variation satisfies an inequality of the form

$\overline{\mathbb{Q}}$31

with $\overline{\mathbb{Q}}$32. On sufficiently small supports this implies strict local concavity, contradicting second-order minimality. Consequently, any local minimizer has vanishing Gauss curvature at every point where curvature is defined. In particular, no globally $\overline{\mathbb{Q}}$33 positively curved symmetric body can be a local minimizer (Harrell et al., 2011).

In dimension $\overline{\mathbb{Q}}$34, the support-function formalism becomes global because convexity is equivalent to the distributional inequality

$\overline{\mathbb{Q}}$35

The paper proves that any symmetric local minimizer must be a polygon, and then refines the second-variation analysis to show that it must in fact be a parallelogram. This recovers and strengthens Mahler’s planar theorem and Reisner’s uniqueness statement for the equality case (Harrell et al., 2011).

5. Powers near integers, Hardy’s question, and Cantor-type approximation

Mahler’s 1957 theorem concerns the fractional parts of powers of algebraic numbers. For rational $\overline{\mathbb{Q}}$36 with $\overline{\mathbb{Q}}$37 and any $\overline{\mathbb{Q}}$38, the set of $\overline{\mathbb{Q}}$39 such that

$\overline{\mathbb{Q}}$40

is finite. Mahler then asked for a classification of real algebraic $\overline{\mathbb{Q}}$41 exhibiting this finiteness property. Corvaja and Zannier proved the exact criterion: there exists some $\overline{\mathbb{Q}}$42 such that

$\overline{\mathbb{Q}}$43

for infinitely many $\overline{\mathbb{Q}}$44 if and only if $\overline{\mathbb{Q}}$45 is a PV number for some positive integer $\overline{\mathbb{Q}}$46. Thus the obstruction is precisely the absence of a Pisot–Vijayaraghavan power (Philippon et al., 2019).

A companion problem of Hardy asks when

$\overline{\mathbb{Q}}$47

can occur. For algebraic $\overline{\mathbb{Q}}$48 and real $\overline{\mathbb{Q}}$49, Hardy had shown that this forces $\overline{\mathbb{Q}}$50 to be a PV number and $\overline{\mathbb{Q}}$51. The later work formulates two sets,

$\overline{\mathbb{Q}}$52

and proves that if $\overline{\mathbb{Q}}$53 with $\overline{\mathbb{Q}}$54 algebraic, then some $\overline{\mathbb{Q}}$55 is a PV number. It further gives exact descriptions of the algebraic pairs in $\overline{\mathbb{Q}}$56 and in $\overline{\mathbb{Q}}$57 in terms of complementary modules and traces (Philippon et al., 2019).

Mahler also asked how well irrational points in the Cantor set can be approximated by rationals in the Cantor set. For a Cantor-like set $\overline{\mathbb{Q}}$58 defined by a digit set $\overline{\mathbb{Q}}$59 with $\overline{\mathbb{Q}}$60, the Hausdorff dimension is

$\overline{\mathbb{Q}}$61

Let $\overline{\mathbb{Q}}$62 be the least integer such that $\overline{\mathbb{Q}}$63 is invariant under $\overline{\mathbb{Q}}$64. Then every $\overline{\mathbb{Q}}$65 admits infinitely many intrinsic rational approximants $\overline{\mathbb{Q}}$66 satisfying

$\overline{\mathbb{Q}}$67

This is the intrinsic analogue of Dirichlet’s theorem, with Hausdorff dimension replacing ambient dimension and a logarithm replacing the Euclidean power law. For the middle-third Cantor set, $\overline{\mathbb{Q}}$68, $\overline{\mathbb{Q}}$69, and $\overline{\mathbb{Q}}$70 (Broderick et al., 2011).

6. Conceptual unity and remaining directions

Despite their diversity, these Mahler problems share a characteristic template: a sparse arithmetic or dynamical structure forces unexpectedly rigid conclusions. In the theory of entire functions, algebraic inputs and outputs can be synchronized across all derivatives (Alves, 2023). In the theory of Mahler equations, simultaneous compatibility with two multiplicatively independent dilations forces rationality, while multivariate admissibility and spectral-radii independence force algebraic independence of values (Adamczewski et al., 2013, Adamczewski et al., 2020, Adamczewski et al., 2018). In Mahler measure, monomial substitutions, random-process limits, and regulator formulas turn a logarithmic torus integral into explicit arithmetic invariants (Brunault et al., 2022, Mossinghoff, 2024, Tao et al., 2022). In convex geometry and Diophantine approximation, local curvature or digit restrictions sharply constrain admissible extremizers and approximation rates (Harrell et al., 2011, Broderick et al., 2011, Philippon et al., 2019).

Several frontier questions remain visible in the cited literature. The derivative version of Problem B yields inclusions rather than equalities for all derivatives (Alves, 2023). The Liouville-number problem is solved for large subclasses $\overline{\mathbb{Q}}$71, not for all Liouville numbers (Marques et al., 2015). Multivariate Mahler theory in positive characteristic and beyond the current admissibility framework remains open (Adamczewski et al., 2020). The supremum of $\overline{\mathbb{Q}}$72 for Littlewood polynomials is not known, even though Turyn polynomials push the lower bound past $\overline{\mathbb{Q}}$73 (Mossinghoff, 2024). The symmetric Mahler conjecture in dimension $\overline{\mathbb{Q}}$74 is unresolved, notwithstanding the curvature obstruction and the full planar classification (Harrell et al., 2011). On Cantor-like sets, the intrinsic Dirichlet theorem is established, but optimal counting laws for rationals in the set—and hence full Khintchine–Jarník analogues—remain conjectural (Broderick et al., 2011).

In that sense, the Mahler problem is best understood not as a single statement but as a research program: arithmetic self-similarity, whether encoded by $\overline{\mathbb{Q}}$75, by exceptional approximation patterns, or by toric logarithmic integrals, repeatedly produces unexpectedly sharp rigidity phenomena.