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on Wirsing Constant ($C_{83}$)

The statement

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.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.