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OP-12098SolvedNumber theory

Wirsing Constant (C83C_{83})

Wirsing Constant: determine C83C_{83} or improve its bounds; the repository records 0.30366300 (lower) and 0.30366300 (upper).

MSC 11K50 · Metric theory of continued fractionsMSC 11Y60 · Evaluation of number-theoretic constants

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

The Gauss--Kuzmin--Wirsing (GKW) operator acts on suitable function spaces on [0,1][0,1] by

(Lf)(x)=∑k=1∞1(x+k)2f ⁣(1x+k).(\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}T(x) = \{1/x\}, which generates the continued fraction expansion. Its spectral radius is 11, attained by the simple dominant eigenvalue λ1=1\lambda_1 = 1 (with eigenfunction proportional to the Gauss measure density (1+x)−1log⁡2(1+x)^{-1}\log 2). The subdominant eigenvalue λ2\lambda_2 is real and negative.

The Wirsing constant C83C_{83} is defined as

C83=∣λ2∣≈0.3036630028987326586…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 nn-th continued-fraction partial quotient from its Gauss-measure limit decays like C83nC_{83}^n as n→∞n\to\infty. The problem of determining C83C_{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.

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Background

  • The second eigenvalue satisfies λ2<0\lambda_2 < 0, so C83=−λ2C_{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 λ~2=−0.30366 30028 98732 65859 74481 21901 55623 31108 77352 25365\tilde\lambda_2 = -0.30366\,30028\,98732\,65859\,74481\,21901\,55623\,31108\,77352\,25365 78951 88245 48146 72269 95294 24691 09843 40811 93436 3636878951\,88245\,48146\,72269\,95294\,24691\,09843\,40811\,93436\,36368 11098 27226 37106 16938 47461 48597 45801 31606 52653 8181811098\,27226\,37106\,16938\,47461\,48597\,45801\,31606\,52653\,81818 23787 91324 46139 89647 6429723787\,91324\,46139\,89647\,64297, with ∣λ2−λ~2∣<10−175\lvert\lambda_2 - \tilde\lambda_2\rvert < 10^{-175}.
  • The GKW operator is compact on Hardy spaces H∞(D)H^\infty(D) for suitable complex discs D⊃[0,1]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 (from the repository page)

  • [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

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.30366300; best known upper bound 0.30366300. Values marked with an asterisk have minimal verification. The recorded bounds coincide, so the repository treats the value as determined.

Known upper bounds

Bound Reference Comments
2/(3+5)≈0.381972/(3+\sqrt{5}) \approx 0.38197 [Schweiger1980], cited in [MR1987] Simple rigorous bound from metrical theory
0.303663270.30366327 [MR1987], eq. (5.20) Certified via generalized Temple inequalities
0.30366300289873265859 ⁣… ⁣6424297+10−1750.30366300289873265859\!\ldots\!6424297 + 10^{-175} [GKW2026] Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits

Known lower bounds

Bound Reference Comments
0.303662990.30366299 [MR1987], eq. (5.20) Certified via generalized Temple inequalities
0.30366300289873265859 ⁣… ⁣6424297−10−1750.30366300289873265859\!\ldots\!6424297 - 10^{-175} [GKW2026] Certified spectral enclosure for the infinite-dimensional GKW operator; 175 digits
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Wirsing Constant ($C_{83}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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