Locally constant logarithmic potentials

Definitions and mathematical statement

Identify the plane with ${\mathbb{C}}$, and write $D_r=\{z\in{\mathbb{C}}:|z|\le r\}$ and $C_r=\{z\in{\mathbb{C}}:|z|=r\}$ for $r>0$. The question concerns a positive Borel measure $\mu$ satisfying the growth condition \[ \exists c>0\ \exists\alpha\in(0,1/2)\ \forall r>0,\qquad \mu(D_r)\le cr^\alpha, \] and its logarithmic potential \[ u(z)=\int_{{\mathbb{C}}}\log\left|1-\frac z\zeta\right|\,\mathrm d\mu(\zeta). \] The all-radii growth interpretation implies $\mu(\{0\})=0$, so the integrand at $\zeta=0$ may be ignored. Local constancy is understood for finite real values of the potential, rather than a constant value $-\infty$.

Does there exist such a measure and an open set $U\subseteq{\mathbb{C}}$ such that \[ \forall r>0,\quad U\cap C_r\ne\varnothing, \] \[ \forall z\in U\ \exists\varepsilon>0,\quad B(z,\varepsilon)\subseteq U\quad\text{and}\quad \forall w\in B(z,\varepsilon),\ u(w)=u(z)\in{\mathbb{R}}? \] Local constancy permits different constants on different components of $U$; no connectedness or single common value is imposed. The source does not explicitly exclude $\mu=0$. If positive means nonnegative, that measure gives a trivial affirmative example. The substantive nonzero-measure interpretation adds $\mu({\mathbb{C}})>0$, and this missing qualification must be checked rather than silently inserted. The source also leaves the radius quantifier implicit; the displayed formulation uses all $r>0$, not merely sufficiently large radii. The supplied research summary reports no located resolution and attributes the problem to Eremenko--Lewis; this is an unverified literature assessment.

Short English statement

Can a nontrivial positive plane measure of growth $O(r^\alpha)$, $0<\alpha<1/2$, have a logarithmic potential locally constant on an open set intersecting every centered circle? Nontriviality and the radius convention need source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $3700020$, exact source code catalogue entry, "Locally constant logarithmic potentials." Registry proposer: Alexandre Eremenko and cited proposers; proposed year 2008.
• Eremenko, My favorite unsolved problems, Potential theory, Problem 1; https://www.math.purdue.edu/~eremenko/uns1.html. Recorded access 2026-07-29; extraction recorded as pdf-text.
• Supplied OPEN-TRIAGE research summary attributes the problem to Eremenko--Lewis and names equivpot.pdf, but supplies no complete publication citation. Its explanation of the exponent threshold is not accompanied by a theorem or reference.