A critical majorant for the Khinchin-Ostrowski property

TL;DR

This paper establishes an optimal majorant condition for the Khinchin-Ostrowski property on the unit circle, removing previous integrability restrictions and providing precise criteria for uniqueness sets based on geometric and harmonic measure estimates.

Contribution

It introduces a new approach using Joukowski-Privalov domains to determine the critical majorant for the Khinchin-Ostrowski property, extending prior results without integrability assumptions.

Findings

Identifies the critical majorant for the property.

Provides explicit harmonic measure estimates via conformal mappings.

Shows the only relevant characteristic is interval containment above the critical majorant.

Abstract

In this short note we prove an optimal version of a classical result. Given a majorant determining a growth restriction on functions in the unit disk $\mathbb{D}$, we say that a set $E$ on the unit circle $\mathbb{T}$ is a uniqueness set, or has the Khinchin-Ostrowski property, with respect to the majorant, if any sequence of analytic polynomials satisfying the growth restriction which converges in an appropriate sense to $0$ on $E$, in fact is forced to converge to $0$ in $\mathbb{D}$ also. Theorems proved by Kegejan and Khrushchev state that if $E$ has positive Lebesgue measure and satisfies a generalized Beurling-Carleson condition, then for an appropriate majorant the Khinchin-Ostrowski property is satisfied. A technical point in Khrushchev's proof is the estimation of the harmonic measure in a Privalov-type domain which requires logarithmic integrability of the majorant. This forbids the application of his result to certain types of generalized Beurling-Carleson conditions. Here, we dispose of the integrability assumption on the majorant. To do so, we use a Joukowski-Privalov domain which is obtained by removing from the unit disk the areas enclosed by hyperbolic geodesics between the endpoints of intervals complementary to $E$. For this type of domain the method of Khrushchev applies, but the harmonic measure may be estimated more accurately by simple explicit formulas for conformal mappings. As a consequence, we find the critical majorant at which Beurling-Carleson type conditions stop determining the Khinchin-Ostrowski property of a set, and above which the containment of intervals is the only relevant characteristic. We discuss also weighted versions of the Khinchin-Ostrowski property, and apply our result to establish the remarkable precision of a one-sided spectral decay condition which detects the local logarithmic integrability of a function.