A critical majorant for the Khinchin-Ostrowski property

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.