Carleson–Jones quarter conjecture

Definitions and mathematical statement

Let $E\subset\mathbb C$ be a connected compact set of positive logarithmic capacity. Let $\Omega_E$ be the unbounded component of its complement. Its Green function with pole at infinity is the positive harmonic function $G_E$ on $\Omega_E$ normalized by \[ G_E(z)=\log|z|-\log\operatorname{cap}(E)+o(1) \quad(z\to\infty). \] Require regularity: $G_E$ extends continuously by zero to the boundary. Extend it by zero also over $E$ and the bounded complementary components.

For $\varepsilon>0$, define the total level-set length and its growth exponent by \[ \ell_E(\varepsilon)= \mathcal H^1\{z:G_E(z)=\varepsilon\}, \qquad \beta_E=\limsup_{\varepsilon\downarrow0} \frac{\log\ell_E(\varepsilon)}{-\log\varepsilon}. \] Length means one-dimensional Hausdorff measure, including all components of the level curve. Put $\beta=\sup_E\beta_E$, over all such regular connected compact sets.

Is $\beta=1/4$? A proof requires a universal upper bound $\beta_E\leq1/4$ and examples whose exponents approach $1/4$ arbitrarily closely. It need not supply a single compact set attaining the supremum. The limiting variable is the Green-function level, rather than Euclidean distance to the compact set. Translation or fixed rescaling of $E$ changes level lengths only by a constant factor and leaves this exponent unchanged. No smoothness or finite boundary length is assumed for $E$ itself.

Short English statement

Is one quarter the largest possible growth exponent for lengths of Green-function level curves around a connected compact set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700033 (AMR-036-0033), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Littlewood constants, Section 2 conjecture. Original source indexed by AMR-036-0033. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Some constants coming from the work of Littlewood, Section 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/lit.pdf.