Separating the Polar of Sets of Uniqueness from the Polars of All $H^{(d)}$-Sets

Abstract

We construct a perfect set of uniqueness on the circle carrying a continuous probability measure which annihilates every $H^{(d)}$-set, simultaneously for all finite dimensions d. This gives a strict inclusion between the polar of the compact sets of uniqueness and the polar of the union of the classes $H^{(d)}$, answering a question raised in earlier work on these polars. The construction uses random binary increments of size comparable to the reciprocal of a product of square roots. A Gaussian sampling estimate gives a uniform Fourier-tail bound for all pseudomeasures supported by the resulting set, and also bounded atomic approximation and spectral synthesis. A separate energy estimate, stable under deletion of a small number of increments, yields full support for every limiting quasi-independent image of every positive measure absolutely continuous with respect to the constructed measure.