Approximation Numbers in Operator and Number Theory

• Approximation numbers are quantitative invariants that measure how closely operators or numbers can be approximated by simpler, low-complexity entities.
• In operator theory, they generalize singular values by quantifying the distance to finite-rank operators, vital for spectral analysis and embedding estimates.
• In Diophantine settings, these numbers gauge the quality of rational and algebraic approximations, linking approximation constants, exact orders, and Hausdorff dimensions.

Approximation numbers are quantitative invariants of approximability. In operator theory, the $k$-th approximation number of a bounded linear operator measures the distance to the class of operators of rank $<k$; in metric Diophantine approximation, the same expression is also used for approximation functions, constants, and exponents that quantify how well real numbers can be approximated by rationals, algebraic numbers, or partial sums arising from symbolic expansions. The common theme is the measurement of finite or discrete approximation, but the ambient structures, invariants, and theorems are different (P, 2024, Granville, 2023, Cheng et al., 29 Jun 2026).

1. Terminological scope

In the cited literature, “approximation numbers” is not a single invariant but a family of related notions. In functional analysis it belongs to the theory of $s$-numbers and quantifies approximation by finite-rank operators. In Diophantine approximation it refers to approximation functions such as $\psi(n)$, to exponents such as $\omega_n^*(\xi)$, and to simultaneous approximation constants such as $\lambda_{k,j}$ and $\widehat{\lambda}_{k,j}$. In Cantor series expansions it is tied to the exact approximation order of the partial sums $\omega_n(x)$ (P, 2024, Poëls, 2024, Schleischitz, 2014, Cheng et al., 29 Jun 2026).

Setting	Object called approximation numbers	Representative expression
Operator theory	Distance to rank $<k$ operators	$$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$
Rational approximation	Approximation function	$<k$0
Algebraic approximation	Exponent of approximation by algebraic numbers of degree $<k$1	$<k$2
Cantor series	Exact $<k$3-approximability by partial sums	$<k$4

A persistent source of confusion is that these quantities are not interchangeable. Operator-theoretic approximation numbers are $<k$5-numbers attached to linear maps, whereas Diophantine approximation numbers are attached to the approximation of scalars or tuples by arithmetic objects. The shared terminology reflects a common quantitative idea rather than a common formal definition.

2. Operator-theoretic definition and basic properties

For a bounded linear operator $<k$6 between normed linear spaces, the $<k$7-th approximation number is

$<k$8

where $<k$9 denotes the bounded linear operators of rank less than $s$0. In Banach spaces this definition generalizes singular values, and on Hilbert spaces the approximation numbers coincide with the singular values for compact operators (P, 2024, Lechner et al., 2016).

The quasi-Banach extension uses the same formula: $s$1 The cited axioms include monotonicity and normalization,

$s$2

quasi-additivity,

$s$3

and submultiplicativity,

$s$4

They also satisfy the rank property $s$5, and if $s$6, then $s$7 (Gerhold, 5 Aug 2025).

These formulas identify approximation numbers as a multiplicative $s$8-number sequence. In this role they quantify how well an operator can be approximated by low-complexity surrogates and provide a scale finer than mere boundedness, yet more flexible than spectral data.

3. Convergence, adjoints, and related $s$9-numbers

A central structural result is a generalized convergence theorem for truncations. If $\psi(n)$0 is a dual space of $\psi(n)$1, $\psi(n)$2, and

$\psi(n)$3

in the weak$\psi(n)$4 topology on $\psi(n)$5, with $\psi(n)$6, then for each $\psi(n)$7,

$\psi(n)$8

The cited formulation removes the separability assumptions present in earlier results and uses nets, subnets, and weak$\psi(n)$9 compactness (P, 2024).

The same paper gives a convergence statement for adjoints: $\omega_n^*(\xi)$0 when $\omega_n^*(\xi)$1 in the weak operator topology. It also formulates the complete symmetry problem through the identity $\omega_n^*(\xi)$2, and proves that, in the dual-codomain setting, approximation numbers are attained: for every $\omega_n^*(\xi)$3 there exists $\omega_n^*(\xi)$4 such that $\omega_n^*(\xi)$5 (P, 2024).

Approximation numbers serve as a reference point for other classical $\omega_n^*(\xi)$6-numbers. The cited relations are

$\omega_n^*(\xi)$7

$\omega_n^*(\xi)$8

together with Carl-type representations of Kolmogorov and Gelfand numbers by approximation numbers. In Hilbert spaces, all $\omega_n^*(\xi)$9-numbers coincide with approximation numbers, so convergence statements for truncations transfer directly to Chang, Weyl, Kolmogorov, and Gelfand numbers (P, 2024).

This suggests a broad principle: approximation numbers are often the most convenient “base” $\lambda_{k,j}$0-number, from which parallel results for neighboring width-type quantities can be derived.

4. Entropy numbers, spectral connections, and Sobolev embeddings

The approximation method for entropy numbers is closely parallel to the finite-section method for approximation numbers. For operators $\lambda_{k,j}$1 with $\lambda_{k,j}$2, strong or weak convergence can imply

$\lambda_{k,j}$3

and, for bounded operators between Hilbert spaces,

$\lambda_{k,j}$4

The cited work presents this as a complete answer to a question of B. Carl in the separable Hilbert setting and notes an extension beyond separability through operator-theoretic arguments (Deepesh et al., 2017).

Approximation numbers and entropy numbers are linked explicitly for periodic Sobolev-type embeddings. For spaces $\lambda_{k,j}$5 with weights $\lambda_{k,j}$6, the embedding into $\lambda_{k,j}$7 satisfies

$\lambda_{k,j}$8

where $\lambda_{k,j}$9. This reduces approximation-number estimates to covering-number estimates in finite-dimensional geometry and yields preasymptotic bounds for isotropic Sobolev, analytic, and Gevrey-type spaces (Kühn et al., 2015).

For anisotropic Sobolev embeddings $\widehat{\lambda}_{k,j}$0, the cited preasymptotic and asymptotic regimes are

$\widehat{\lambda}_{k,j}$1

with

$\widehat{\lambda}_{k,j}$2

The same source states that these embedding problems are intractable and do not suffer from the curse of dimensionality (Chen et al., 2016).

For non-periodic Sobolev embeddings on a bounded domain $\widehat{\lambda}_{k,j}$3, approximation numbers are tied directly to elliptic eigenvalue problems through

$\widehat{\lambda}_{k,j}$4

where $\widehat{\lambda}_{k,j}$5 is the $\widehat{\lambda}_{k,j}$6-th eigenvalue of the corresponding Dirichlet or Neumann operator. The resulting bounds are asymptotically sharp and have explicit dependence on $\widehat{\lambda}_{k,j}$7, $\widehat{\lambda}_{k,j}$8, $\widehat{\lambda}_{k,j}$9, and $\omega_n(x)$0 (Mieth, 2018).

Spectral theory supplies an additional interpretation. On Hilbert spaces,

$\omega_n(x)$1

and for self-adjoint compact operators,

$\omega_n(x)$2

These identities and inequalities show that approximation numbers control eigenvalue decay and can coincide with it in the self-adjoint setting (Gerhold, 5 Aug 2025).

5. Composition operators and boundary geometry

For composition operators on analytic function spaces, approximation numbers encode the geometry of boundary contact. On weighted Bergman spaces $\omega_n(x)$3, any compact composition operator satisfies

$\omega_n(x)$4

for suitable $\omega_n(x)$5 and $\omega_n(x)$6. A sharper invariant is

$\omega_n(x)$7

with

$\omega_n(x)$8

Exponential decay is attained only when $\omega_n(x)$9, while arbitrarily slow decay is also possible through explicit compact symbols (Li et al., 2011).

On the Dirichlet space, a parallel phenomenon holds. There exist compact composition operators with

$<k$0

for any sequence $<k$1, so the decay can be arbitrarily close to sub-exponential. At the same time, the cited lower bounds exclude decay faster than exponential, and the geometry of boundary approach is captured by

$<k$2

where $<k$3 and $<k$4. The set of contact points with the unit circle can be any compact set of logarithmic capacity zero, including a singleton (Lefèvre et al., 2012).

For composition operators on $<k$5, finite-dimensional model subspaces provide a general method. When the image domain touches the unit circle at one point, the cited results distinguish several regimes. Smooth tangency permits arbitrarily slow decay and even explicit construction of operators with prescribed slow decay; cusp-type contact yields

$<k$6

while corner-type behavior gives intermediate exponential bounds for lens maps (Queffélec et al., 2013).

Weighted composition operators $<k$7 interpolate between symbol geometry and weight vanishing. The cited lower bounds state that no non-zero weighted composition operator has superexponential decay, and if $<k$8, then for any weight $<k$9,

$$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$0

The precise exponential rate is expressed through Green capacity: $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$1 For weighted lens maps, the upper and lower bounds match in type and yield $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$2 for some $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$3 (Lechner et al., 2016).

Recent work extends the subject to differences of composition operators $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$4 on $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$5 and on $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$6. Lower bounds are obtained from interpolating sequences and Carleson measures; upper bounds are obtained from Blaschke products and pseudohyperbolic control. For smooth perturbations of the form $$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$7, the cited bounds take the form

$$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$8

while corner-type examples give

$$a_k(T) = \inf\left\{ \|T - F\| : F \in \mathcal{F}_k(X,Y) \right\}$$9

Tensor-product reductions then transfer one-variable decay information to bidisc examples (Bayart et al., 24 Sep 2025).

6. Diophantine approximation, approximation constants, and exact order

In metric Diophantine approximation, approximation numbers are attached to arithmetic approximation rather than to operators. A basic rational-approximation model studies the set of $<k$00 for which

$<k$01

for infinitely many pairs $<k$02, and its reduced-fraction analogue $<k$03. Khinchin’s theorem gives the measure criterion for decreasing $<k$04, while the Duffin–Schaeffer conjecture, proved by Koukoulopoulos and Maynard, replaces $<k$05 by the weighted sum

$<k$06

and yields the exact zero-one law for reduced fractions (Granville, 2023).

A closely related irrationality criterion is the existence of “nice rational approximations”: $<k$07 The cited survey states that a real number $<k$08 is irrational if and only if there exists a sequence of coprime integers $<k$09 with $<k$10 satisfying that condition. It also records explicit constructions for $<k$11, $<k$12, $<k$13, $<k$14, and trigonometric values such as $<k$15 and $<k$16 (Bhattacharyya et al., 2022).

Approximation by algebraic numbers of bounded degree is encoded by the exponent

$<k$17

For transcendental $<k$18, Wirsing proved $<k$19; the cited paper improves this to

$<k$20

for each $<k$21, using an approach partly inspired by parametric geometry of numbers (Poëls, 2024).

For simultaneous approximation of power tuples $<k$22, Liouville numbers occupy an extreme position. For any Liouville number $<k$23,

$<k$24

and

$<k$25

For a specific explicit class including $<k$26 and Liouville numbers in the Cantor set, all these values are realized simultaneously for every $<k$27 (Schleischitz, 2014).

The digital-structure viewpoint leads further. For a class of special Liouville numbers constructed from $<k$28-adic expansions or from rapidly increasing divisibility chains $<k$29, the cited work gives explicit formulas for $<k$30 and $<k$31 and shows that one can construct tuples $<k$32 with prescribed simultaneous approximation properties. This ties approximation constants directly to gaps in digit positions in $<k$33-adic expansions (Schleischitz, 2013).

Cantor series expansions provide a symbolic analogue of exact approximation order. For

$<k$34

the limsup set

$<k$35

and the exact set

$<k$36

mirror classical exact approximation by rationals. Under the condition

$<k$37

with $<k$38 non-increasing and $<k$39, the Hausdorff dimensions satisfy

$<k$40

This is presented as a Cantor-series analogue of exact metric Diophantine approximation (Cheng et al., 29 Jun 2026).

Across these settings, the phrase “approximation numbers” names a collection of precise numerical gauges rather than a single invariant. In operator theory it quantifies finite-rank approximability and interfaces with entropy, widths, and spectral decay; in Diophantine approximation it quantifies rational, algebraic, and symbolic approximation, often through zero-one laws, exact-order sets, and Hausdorff dimension formulas. The unifying content is quantitative approximation, but the objects being approximated, the admissible approximants, and the governing theorems are specific to each domain.