Liouville-Like Numbers: Approximations & Applications

• Liouville-like numbers are transcendental numbers characterized by exceptional rational and algebraic approximability, extending the classical Liouville definition.
• Refinements such as strong, ultra-strong, v-Liouville, and L-numbers impose specific growth, spacing, and combinatorial constraints on the approximants.
• These numbers form intricate sets and fields with structured algebraic properties, impacting theories on dynamical mappings and transcendence.

Liouville-like numbers are transcendental real or complex numbers organized around exceptionally strong rational or algebraic approximation, with the classical Liouville numbers as the prototype. In current usage, the phrase does not designate a single canonical class; rather, it encompasses several refinements that control the rate, spacing, degree, or algebraic organization of approximants, as well as constructions that impose additional combinatorial or dynamical constraints such as normality, prescribed finite-state dimension, or stability under nonlinear maps (Kumar et al., 2013, Marques et al., 2023, Bilu et al., 9 Apr 2026, Nandakumar et al., 2012).

1. Classical Liouville numbers and the $U_1$ paradigm

A real number $x$ is a Liouville number if and only if for every $n\in\mathbb{N}$ there exist integers $p,q$ with $q\ge 2$ such that

$\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$

Equivalently, the irrationality exponent $\mu(x)$ is infinite. Liouville’s theorem shows that algebraic irrational numbers cannot satisfy approximation inequalities of this strength, so every Liouville number is transcendental. The set of Liouville numbers has Lebesgue measure zero, classical Hausdorff dimension zero, and constructive Hausdorff dimension zero (Nandakumar et al., 2012).

In Mahler’s and LeVeque’s higher-degree framework, Liouville numbers are exactly the $U_1$-numbers. More generally, a complex transcendental number $\xi$ is a $U_m$-number if $x$0 is the least degree for which there exist infinitely many algebraic numbers $x$1 of degree $x$2 with approximation

$x$3

for arbitrarily large $x$4; in the rational case $x$5, this is precisely the Liouville condition rewritten in height language (Bilu et al., 26 May 2025). This places classical Liouville numbers at the bottom level of a broader hierarchy of “Liouville-like” behavior, where rational approximation is replaced by approximation by algebraic numbers of prescribed degree (Chalebgwa et al., 2022).

The literature also emphasizes that transcendence alone does not determine finer distributional or approximation properties. Liouville numbers are “extremely well approximable,” but they can still display widely differing behaviors with respect to finite-state randomness, normality, exact-degree algebraic approximation, or stability under analytic maps (Nandakumar et al., 2012, Ooto, 2018).

2. Controlled approximation schedules and specialized subclasses

Several refinements strengthen the bare condition $x$6 by prescribing how exceptionally good approximants must occur. One such refinement is the class of strong Liouville numbers, defined via convergents $x$7 by the eventual inequalities $x$8 for every $x$9, equivalently $n\in\mathbb{N}$0 when $n\in\mathbb{N}$1. An even stronger notion is the ultra-strong Liouville condition, where the convergents satisfy

$n\in\mathbb{N}$2

These classes were introduced precisely to obtain robust mapping properties under transcendental entire functions (Chaves et al., 2019, Lelis et al., 2016).

A different direction prescribes the growth scale of rational approximation. The $n\in\mathbb{N}$3-Liouville classes $n\in\mathbb{N}$4 quantify the speed of approximation through inequalities of the form

$n\in\mathbb{N}$5

with $n\in\mathbb{N}$6 governing the growth of $n\in\mathbb{N}$7. In this notation, $n\in\mathbb{N}$8 is the full set of Liouville numbers, $n\in\mathbb{N}$9 for every $p,q$0, and $p,q$1 is $p,q$2-dense; ultra-Liouville numbers form a particularly thin but topologically large subclass defined by

$p,q$3

for every $p,q$4 and infinitely many $p,q$5 (Marques et al., 2023).

Another parametrized family is given by the classes $p,q$6 and $p,q$7, where $p,q$8 is a nondecreasing function with $p,q$9. These consist of Liouville numbers for which approximation exponents at least $q\ge 2$0 can be achieved with denominators $q\ge 2$1, either for every $q\ge 2$2 or eventually for all $q\ge 2$3. This packages not just the existence of good approximants but the speed with which they can be found; for sufficiently rapidly growing $q\ge 2$4, these classes are nonempty, dense, and uncountable in every interval (Marques et al., 2015).

Bilu and Marques introduced $q\ge 2$5-numbers as a refinement adapted to higher-degree algebraic images. A real number $q\ge 2$6 is an $q\ge 2$7-number if there exist rationals $q\ge 2$8 and positive reals $q\ge 2$9 such that

$\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$0

with $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$1. The extra height-growth condition expresses that very good rational approximations are “not too sparse.” Strong Liouville numbers are $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$2-numbers, but the class is wider and includes $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$3 (Bilu et al., 9 Apr 2026).

3. Liouville sets, Liouville fields, and algebraic structure

A major structural generalization replaces individual numbers by families sharing a common approximation architecture. For an increasing sequence $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$4 and a sequence $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$5 with $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$6, define

$\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$7

Such sets are called Liouville sets, and they extend Maillet’s earlier viewpoint by organizing many Liouville numbers around one denominator sequence rather than building one number at a time (Kumar et al., 2013).

The same work defines a Liouville field as a field generated by a Liouville set. In the canonical case $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$8, the closure phenomenon is especially strong: $\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$9 is already a field, denoted $\mu(x)$0. It is closed under addition, subtraction, multiplication, and inversion, and every irrational element of $\mu(x)$1 again lies in $\mu(x)$2. Thus $\mu(x)$3 contains no irrational algebraic numbers; it consists exactly of $\mu(x)$4 together with a controlled family of Liouville numbers (Kumar et al., 2013).

This framework yields several equivalences. A real number $\mu(x)$5 is Liouville if and only if it belongs to some Liouville set, if and only if $\mu(x)$6 itself is a Liouville set for suitable $\mu(x)$7 and $\mu(x)$8, and if and only if $\mu(x)$9 is a Liouville field. Moreover, every Liouville number lies in a continuum-sized Liouville set $U_1$0 such that $U_1$1 is a Liouville field (Kumar et al., 2013).

The associated sets can be topologically large without being structurally trivial. When $U_1$2 is nonempty, it has cardinality continuum and is dense in $U_1$3, but it is not a $U_1$4 dense subset. The paper also constructs disjoint families $U_1$5, shows that subsequence refinement can strictly enlarge a Liouville set, and proves that intersections can behave subtly. This suggests that “Liouville-like” structure is not exhausted by the classical set $U_1$6 itself; it can be stratified by denominator schedules and field-theoretic closure properties (Kumar et al., 2013).

4. Higher-degree analogues and exact-degree approximation

One central theme in the modern theory is the production of $U_1$7-numbers from Liouville-type inputs. Generalized Liouville series provide an explicit mechanism. A strongly lacunary series

$U_1$8

is called a generalized Liouville series if its zero blocks are very long and the quotients $U_1$9 remain bounded. For an algebraic $\xi$0 with $\xi$1, the truncations $\xi$2 furnish algebraic approximants of controlled height, and the main theorem identifies the exact $\xi$3-class of $\xi$4: if $\xi$5 is the smallest degree occurring infinitely often among the algebraic numbers $\xi$6, then $\xi$7 (Bilu et al., 26 May 2025).

The same paper gives an application to simple algebraic integers. If $\xi$8 is a simple algebraic integer of degree $\xi$9, $U_m$0, and the block values $U_m$1 are nonzero infinitely often, then $U_m$2. For the classical series $U_m$3, this yields explicit $U_m$4-values at radicals such as $U_m$5 under the stated hypotheses (Bilu et al., 26 May 2025).

A complementary theorem of Bilu and Marques studies algebraic functions $U_m$6 of degree $U_m$7. If $U_m$8 has genus $U_m$9 and Galois group $x$00 or $x$01, then for any $x$02-number $x$03 in the domain of $x$04,

$x$05

The same article proves that, under suitable simple-zero hypotheses on a polynomial $x$06, the value $x$07 is a $x$08-number for every $x$09-number $x$10; and if $x$11 is a non-real algebraic number of degree $x$12 with Galois group $x$13, then $x$14 for every Liouville number $x$15 (Bilu et al., 9 Apr 2026).

Rational maps with coefficients in a number field of degree $x$16 provide another route from rational to algebraic approximation. Chaves, Marques, and Trojovský define a class $x$17 of Liouville numbers admitting rational approximants $x$18 with

$x$19

where $x$20. For any irreducible rational function $x$21 with $x$22, they show $x$23. The class $x$24 strictly contains the strong Liouville numbers and includes the classical Liouville constant $x$25 (Chaves et al., 2019).

Exact-degree exponents reveal that strong Liouville behavior is not generic inside $x$26. Ooto’s note states that if $x$27 is a strong Liouville number, then for every integer $x$28,

$x$29

By contrast, he constructs quasi-periodic continued fractions $x$30 that are Liouville but not strong and satisfy

$x$31

There are uncountably many such examples. This shows that the exact-degree quadratic approximation spectrum of general Liouville numbers can be radically larger than in the strong class (Ooto, 2018).

5. Distributional and combinatorial refinements

Liouville-like behavior also interacts with distributional notions usually associated with randomness. A striking example is normality. For each base $x$32, there exists a Liouville number normal in base $x$33, obtained by concatenating de Bruijn sequences $x$34 of increasing order with rapidly growing multiplicities: $x$35 This construction is purely combinatorial. The same paper proves that for every rational $x$36 there exists a Liouville number of finite-state dimension $x$37, showing that the measure-zero set of Liouville numbers carries a full rational spectrum of finite-state randomness (Nandakumar et al., 2012).

The finite-state dimension result refines Staiger’s theorem that the set of Liouville numbers has constructive Hausdorff dimension zero. There is no contradiction: constructive Hausdorff dimension is a stronger, algorithmically sensitive notion, whereas finite-state dimension measures only what finite-memory devices detect. In the same work, a number-theoretic construction using primitive roots gives, conditional on a generalized Artin conjecture, Liouville numbers simultaneously normal in any prescribed finite set of bases (Nandakumar et al., 2012).

Another quantitative axis concerns simultaneous approximation of powers. For a Liouville number $x$38, Schleischitz determines the approximation constants for $x$39. He proves that for every $x$40,

$x$41

and

$x$42

For a special class $x$43 with divisibility $x$44 and sufficiently rapid growth, the non-uniform constants attain the exact extremal values

$x$45

simultaneously for all $x$46 and $x$47 (Schleischitz, 2014).

A related $x$48-adic approach constructs tuples $x$49 with prescribed simultaneous approximation spectra by synchronizing long zero blocks in their base-$x$50 expansions. The resulting formulas express $x$51 and $x$52 through growth rates of the mixed digit-position sequence $x$53, and explicit choices recover Liouville and Liouville-like examples, including constructions in the Cantor set (Schleischitz, 2013). Taken together, these results show that extreme rational approximability can coexist with highly structured combinatorial behavior rather than forcing a unique randomness profile.

6. Functional images, self-powers, and transcendence phenomena

Maillet’s classical theorem states that every non-constant rational function with rational coefficients maps Liouville numbers to Liouville numbers. This remains the baseline closure result for the classical class $x$54, and it motivates Mahler’s question whether there exists a transcendental entire function $x$55 such that $x$56. That question remains open (Lelis et al., 2016, Marques et al., 2015).

Partial positive results are known for large subclasses. Kumar, Thangadurai, and Waldschmidt construct an uncountable class $x$57 of ultra-strong Liouville numbers and a broad family of transcendental entire functions

$x$58

with ultra-lacunary support such that $x$59. Marques and Ramirez, in a different direction, define $x$60-ultra numbers—a dense $x$61 subclass of $x$62-numbers of type $x$63—and prove the existence of uncountably many analytic transcendental functions $x$64 with $x$65 (Lelis et al., 2016, Marques et al., 2014).

Marques and Schleischitz enlarge the source classes further by using the parametrized families $x$66. For every $x$67, they construct uncountably many entire transcendental functions $x$68, with $x$69, such that for every derivative order $x$70,

$x$71

The same work proves a sharp dichotomy for power maps: on large subsets of $x$72, integer powers preserve Liouville-ness, whereas fractional powers $x$73 with $x$74 need not (Marques et al., 2015).

The arithmetic of elementary functions at Liouville and $x$75-numbers is much more rigid than Maillet-type preservation might suggest. If $x$76 is a Liouville number, then

$x$77

and all inverse trigonometric and inverse hyperbolic branch values at $x$78, where defined, are transcendental; the same holds for all $x$79-numbers (Chalebgwa et al., 2022). At the opposite end, values of $x$80-functions at algebraic points are never Liouville numbers, because their irrationality exponents are finite (Fischler et al., 2023).

Self-powers and towers exhibit both Liouville and non-Liouville behavior. For the classes $x$81, if $x$82, then

$x$83

is transcendental for $x$84, and in particular there is an explicit $x$85-dense set $x$86 on which $x$87 is transcendental (Marques et al., 2023). At the same time, recent Cantor-type constructions produce a perfect set $x$88 of cardinality $x$89 such that $x$90 for every $x$91, and an even smaller perfect set $x$92 with $x$93 for all $x$94. These results show that Liouville-like approximation can be engineered to survive sums, products, self-powers, and pairwise powers on large perfect sets, while other strengthened Liouville classes force transcendental but non-Liouville self-powers (Morris et al., 21 Nov 2025).