--- title: Approximation Numbers in Operator and Number Theory url: https://www.emergentmind.com/topics/approximation-numbers type: topic --- # Approximation Numbers in Operator and Number Theory 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 \( Cd, \end{cases} \] with \[ g(\mathbf{R}) = \left( \sum_{j=1}^d \frac{1}{R_j} \right)^{-1}. \] The same source states that these embedding problems are intractable and do not suffer from the curse of dimensionality [1607.01865]. For non-periodic Sobolev embeddings on a bounded domain \(\Omega\subset\mathbb{R}^d\), approximation numbers are tied directly to elliptic eigenvalue problems through \[ a_k(\mathrm{id})=\lambda_k^{-1/2}, \] where \(\lambda_k\) is the \(k\)-th eigenvalue of the corresponding Dirichlet or Neumann operator. The resulting bounds are asymptotically sharp and have explicit dependence on \(m\), \(d\), \(\operatorname{vol}\Omega\), and \(\operatorname{vol}A_m\) [1811.01576]. Spectral theory supplies an additional interpretation. On Hilbert spaces, \[ \prod_{i=1}^n |\lambda_i(T)| \leq \prod_{i=1}^n a_i(T), \qquad \sum_{i=1}^n |\lambda_i(T)|^p \leq \sum_{i=1}^n [a_i(T)]^p, \] and for self-adjoint compact operators, \[ |\lambda_n(T)|=a_n(T). \] These identities and inequalities show that approximation numbers control eigenvalue decay and can coincide with it in the self-adjoint setting [2508.06542]. ## 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 \(\mathfrak{B}_\alpha\), any compact composition operator satisfies \[ a_n(C_\varphi)\ge c\,r^n \qquad (n\ge 1) \] for suitable \(c>0\) and \(01-h}|\varphi'(z)|^2\,dA(z)\). The set of contact points with the unit circle can be any compact set of logarithmic capacity zero, including a singleton [1212.4366]. For composition operators on \(\mathcal H^2\), 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 \[ a_n(C_{\varphi_U}) \asymp \exp\left(-\left(\frac{\pi^2}{2}+o(1)\right)\frac{n}{w_U(n)}\right), \] while corner-type behavior gives intermediate exponential bounds for lens maps [1302.4116]. Weighted composition operators \(T=M_wC_\varphi\) interpolate between symbol geometry and weight vanishing. The cited lower bounds state that no non-zero weighted composition operator has superexponential decay, and if \(\|\varphi\|_\infty=1\), then for any weight \(w\), \[ a_n(T)\gtrsim \exp(-n\epsilon_n), \qquad \epsilon_n\to 0. \] The precise exponential rate is expressed through Green capacity: \[ \beta(T):=\lim_{n\to\infty}[a_n(T)]^{1/n} =\exp\left(-\frac{1}{\mathrm{Cap}(\varphi(\mathbb D))}\right). \] For weighted lens maps, the upper and lower bounds match in type and yield \(a_n(T)\asymp n^{-\alpha}\) for some \(\alpha>0\) [1612.01177]. Recent work extends the subject to differences of composition operators \(C_\varphi-C_\psi\) on \(H^2(\mathbb D)\) and on \(H^2(\mathbb D^2)\). 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 \(\psi(z)=\varphi(z)+O((z-1)^\alpha)\), the cited bounds take the form \[ \frac{1}{(\log n)n^{\alpha-2}} \lesssim a_n(C_\varphi-C_\psi) \lesssim \left(\frac{\log n}{n}\right)^{\alpha-2}, \] while corner-type examples give \[ \exp\left(-a\frac{n}{\log n}\right)\lesssim a_n(C_\varphi-C_\psi) \lesssim \exp\left(-b\frac{n}{\log n}\right). \] Tensor-product reductions then transfer one-variable decay information to bidisc examples [2509.19797]. ## 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 \(\alpha\in[0,1)\) for which \[ \left|\alpha-\frac{m}{n}\right|\le \frac{\psi(n)}{n^2} \] for infinitely many pairs \((m,n)\), and its reduced-fraction analogue \(\mathcal L^*(\psi)\). Khinchin’s theorem gives the measure criterion for decreasing \(\psi\), while the Duffin–Schaeffer conjecture, proved by Koukoulopoulos and Maynard, replaces \(\sum \psi(n)/n\) by the weighted sum \[ \sum_{n\ge 1}\frac{\varphi(n)}{n}\frac{\psi(n)}{n} \] and yields the exact zero-one law for reduced fractions [2308.03126]. A closely related irrationality criterion is the existence of “nice rational approximations”: \[ 0<|q_n\alpha-p_n|\to 0. \] The cited survey states that a real number \(\alpha\) is irrational if and only if there exists a sequence of coprime integers \(p_n,q_n\) with \(q_n>0\) satisfying that condition. It also records explicit constructions for \(\sqrt m\), \(e\), \(e^k\), \(e^{p/q}\), and trigonometric values such as \(\sin(1/m)\) and \(\cos(1/m)\) [2206.12579]. Approximation by algebraic numbers of bounded degree is encoded by the exponent \[ \omega_n^*(\xi)=\sup\left\{\omega>0 \;\middle|\; \text{there exist infinitely many algebraic numbers } \alpha \text{ with } \deg(\alpha)\le n : 0<|\xi-\alpha|\le H(\alpha)^{-\omega-1}\right\}. \] For transcendental \(\xi\), Wirsing proved \(\omega_n^*(\xi)\ge (n+1)/2\); the cited paper improves this to \[ \omega_n^*(\xi)\ge \frac{n}{2-\log 2} \] for each \(n\ge 2\), using an approach partly inspired by parametric geometry of numbers [2405.08341]. For simultaneous approximation of power tuples \((\zeta,\zeta^2,\ldots,\zeta^k)\), Liouville numbers occupy an extreme position. For any Liouville number \(\zeta\), \[ \lambda_{k,1}(\underline{\zeta})=\infty,\qquad \lambda_{k,j}(\underline{\zeta})=\frac{1}{j-1}\ \ (2\le j\le k+1), \] and \[ \widehat{\lambda}_{k,1}(\underline{\zeta})=\frac{1}{k},\qquad \widehat{\lambda}_{k,j}(\underline{\zeta})=0\ \ (2\le j\le k+1). \] For a specific explicit class including \(\sum_{n\ge1}10^{-n!}\) and Liouville numbers in the Cantor set, all these values are realized simultaneously for every \(k\ge 1\) [1409.1396]. The digital-structure viewpoint leads further. For a class of special Liouville numbers constructed from \(s\)-adic expansions or from rapidly increasing divisibility chains \(q_{n,j}\), the cited work gives explicit formulas for \(\omega_j\) and \(\widehat\omega_j\) and shows that one can construct tuples \((\zeta_1,\ldots,\zeta_k)\) with prescribed simultaneous approximation properties. This ties approximation constants directly to gaps in digit positions in \(s\)-adic expansions [1301.2177]. Cantor series expansions provide a symbolic analogue of exact approximation order. For \[ x=\frac{\varepsilon_1(x)}{q_1}+\frac{\varepsilon_2(x)}{q_1q_2}+\cdots, \qquad \omega_n(x)=\frac{\varepsilon_1(x)}{q_1}+\cdots+\frac{\varepsilon_n(x)}{q_1\cdots q_n}, \] the limsup set \[ W_c(\psi):=\left\{x\in[0,1): x-\omega_n(x)<\frac{\psi(n)}{q_1\cdots q_n}\ \text{for infinitely many }n\right\} \] and the exact set \[ E_c(\psi):=W_c(\psi)\setminus\bigcup_{0