The Wirsing Constant
Description of constant
The Gauss–Kuzmin–Wirsing (GKW) operator acts on suitable function spaces on $[0,1]$ by
\[(\mathcal{L} f)(x) = \sum_{k=1}^\infty \frac{1}{(x+k)^2} f\!\left(\frac{1}{x+k}\right).\]This is the transfer operator of the Gauss map $T(x) = {1/x}$, which generates the continued fraction expansion. Its spectral radius is $1$, attained by the simple dominant eigenvalue $\lambda_1 = 1$ (with eigenfunction proportional to the Gauss measure density $(1+x)^{-1}\log 2$). The subdominant eigenvalue $\lambda_2$ is real and negative.
The Wirsing constant $C_{83}$ is defined as
\[C\_{83} = \lvert \lambda\_2 \rvert \approx 0.3036630028987326586\ldots\]It controls the rate of convergence of the Gauss–Kuzmin distribution: the deviation of the distribution of the $n$-th continued-fraction partial quotient from its Gauss-measure limit decays like $C_{83}^n$ as $n\to\infty$. The problem of determining $C_{83}$ to high precision appears as Exercise 22 of Section 4.5.3 in the first edition of Knuth’s The Art of Computer Programming (attributed to Gauss), and was reformulated in spectral terms (attributed to Babenko) in the third edition.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $2/(3+\sqrt{5}) \approx 0.38197$ | [Schweiger1980], cited in [MR1987] | Simple rigorous bound from metrical theory |
| $0.30366327$ | [MR1987], eq. (5.20) | Certified via generalized Temple inequalities |
| $0.30366300289873265859!\ldots!6424297 + 10^{-175}$ | [GKW2026] | Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0.30366299$ | [MR1987], eq. (5.20) | Certified via generalized Temple inequalities |
| $0.30366300289873265859!\ldots!6424297 - 10^{-175}$ | [GKW2026] | Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits |
Additional comments and links
- The second eigenvalue satisfies $\lambda_2 < 0$, so $C_{83} = -\lambda_2$.
- Prior to [GKW2026], high-precision non-certified numerical estimates were widely available: Wirsing [Wirsing1974] gave 20 digits (with the caveat, noted in [MR1987], that it was unclear how many could be trusted); MacLeod [MacLeod1993] gave approximately 14 digits via Chebyshev extrapolation; Flajolet–Vallée [FV1994] gave approximately 30 digits; Briggs [Briggs2003] computed 385 digits; and by 2012, over 480 digits had been computed non-rigorously [Alkauskas2012].
- [GKW2026] also certifies the next 49 eigenvalues of the GKW operator, each to at least 90 decimal digits, together with the associated eigenvectors and Riesz spectral projectors.
- The full 175-digit center value is $\tilde\lambda_2 = -0.30366\,30028\,98732\,65859\,74481\,21901\,55623\,31108\,77352\,25365$ $78951\,88245\,48146\,72269\,95294\,24691\,09843\,40811\,93436\,36368$ $11098\,27226\,37106\,16938\,47461\,48597\,45801\,31606\,52653\,81818$ $23787\,91324\,46139\,89647\,64297$, with $\lvert\lambda_2 - \tilde\lambda_2\rvert < 10^{-175}$.
- The GKW operator is compact on Hardy spaces $H^\infty(D)$ for suitable complex discs $D \supset [0,1]$; the full spectrum is discrete and real.
- Wikipedia page on the Gauss–Kuzmin–Wirsing operator
- Code: orkolorko/GKWExperiments.jl; Data: Harvard Dataverse, doi:10.7910/DVN/HKM3Y2
References
- [Alkauskas2012] Alkauskas, Giedrius. Transfer operator for the Gauss’ continued fraction map. I. Structure of the eigenvalues and trace formulas. arXiv:1210.4083 (2012/2018).
- [Briggs2003] Briggs, Keith. A precise computation of the Gauss–Kuzmin–Wirsing constant. Unpublished (2003). Available at http://keithbriggs.info/documents/wirsing.pdf.
- [FV1994] Flajolet, Philippe and Vallée, Brigitte. Continued fraction algorithms, functional operators, and structure constants. Theoret. Comput. Sci. 194 (1-2) (1998), 1–34.
- [GKW2026] Nisoli, Isaia. Certified spectral approximation of transfer operators and the Gauss map. Preprint (2026). arXiv:2602.19435.
- [MacLeod1993] MacLeod, A. J. High-accuracy numerical values in the Gauss–Kuz’min continued fraction problem. Comput. Math. Appl. 26 (3) (1993), 37–44.
- [MR1987] Mayer, Dieter and Roepstorff, Gert. On the relaxation time of Gauss’s continued-fraction map. I. The Hilbert space approach (Koopmanism). J. Statist. Phys. 47 (1–2) (1987), 149–171.
- [Schweiger1980] Schweiger, Fritz. The metrical theory of the Jacobi–Perron algorithm. Lecture Notes in Mathematics, No. 334. Springer, Berlin, 1980.
- [Wirsing1974] Wirsing, Eduard. On the theorem of Gauss–Kuzmin–Lévy and a Frobenius-type theorem for function spaces. Acta Arith. 24 (1974), 507–528. doi:10.4064/aa-24-5-507-528
Contribution notes
Claude Code (claude-sonnet-4-6) was used to assist in researching the historical bounds and preparing this submission. All references and mathematical content were reviewed and verified by the author.