Characterization of measures satisfying all integer-polynomial inequalities

Determine all compactly supported probability measures on a compact subset of the complex plane that satisfy the inequalities \(\int \log |Q(z)|\,d\mu(z)\ge 0\) for every nonzero integer polynomial \(Q\in\mathbb{Z}[z]\).

Background

Arithmetic probability measures are characterized by the requirement that their logarithmic integrals against every nonzero integer polynomial be nonnegative. Although this gives a complete theoretical characterization, it involves infinitely many inequalities and does not provide a practical description of which measures satisfy them.

The paper develops finite criteria for several special families of measures, including convex combinations of equilibrium and harmonic measures, and obtains complete reductions for certain compact sets such as intervals with integer endpoints. The broader problem of characterizing all compactly supported measures satisfying the full infinite system remains unresolved and is posed as the Sarnak question.

References

As a result, it is not well understood which measures, even on circles and intervals, satisfy all these inequalities simultaneously. This motivates the following more general question, attributed to Peter Sarnak in page~2. What are the measures with compact support $K$ that satisfy the infinite number of inequalities in E:infinite?

E:infinite:

∫log⁡∣Q(z)∣ dμ(z)≥0∀ Q∈Z[z]∖{0},\int \log |Q(z)|\, d\mu(z) \geq 0 \qquad \forall\, Q \in Z[z] \setminus \{0\},

— Arithmetic probability measures  (2609.03939 - Londhe, 3 Sep 2026) in Question 1 (Sarnak), Section 1, Introduction