From 2985a56fcb41f49127c688742f213e70ec43df47 Mon Sep 17 00:00:00 2001 From: Isaia Nisoli Date: Mon, 23 Mar 2026 16:01:21 -0300 Subject: [PATCH 1/2] Add Wirsing constant (C_83): GKW operator second eigenvalue Certified to 175 decimal digits in arXiv:2602.19435, improving on the 6-digit certified enclosure of Mayer--Roepstorff (1987). Co-Authored-By: Claude Sonnet 4.6 --- constants/83a.md | 71 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 71 insertions(+) create mode 100644 constants/83a.md diff --git a/constants/83a.md b/constants/83a.md new file mode 100644 index 0000000..01e8bfe --- /dev/null +++ b/constants/83a.md @@ -0,0 +1,71 @@ +# 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 via interval arithmetic; 175-digit enclosure | + +## 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 via interval arithmetic; 175-digit enclosure | + +## 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](https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin%E2%80%93Wirsing_operator) +- Code: [orkolorko/GKWExperiments.jl](https://github.com/orkolorko/GKWExperiments.jl); Data: [Harvard Dataverse, doi:10.7910/DVN/HKM3Y2](https://doi.org/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](https://arxiv.org/abs/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](https://doi.org/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. From e19945db63770b5d61ddde614ccd79d314f83786 Mon Sep 17 00:00:00 2001 From: Isaia Nisoli Date: Mon, 23 Mar 2026 16:05:03 -0300 Subject: [PATCH 2/2] Improve description: certified spectral enclosure, not just interval arithmetic Co-Authored-By: Claude Sonnet 4.6 --- constants/83a.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/constants/83a.md b/constants/83a.md index 01e8bfe..0f24d40 100644 --- a/constants/83a.md +++ b/constants/83a.md @@ -26,14 +26,14 @@ terms (attributed to Babenko) in the third edition. | ----- | --------- | -------- | | $2/(3+\sqrt{5}) \approx 0.38197$ | [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 via interval arithmetic; 175-digit enclosure | +| $0.30366300289873265859\!\ldots\!6424297 + 10^{-175}$ | [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 via interval arithmetic; 175-digit enclosure | +| $0.30366300289873265859\!\ldots\!6424297 - 10^{-175}$ | [GKW2026] | Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits | ## Additional comments and links