Brown-Shields Conjecture

The Brown-Shields conjecture asserts that a function in the classical Dirichlet space is a cyclic vector for multiplication by iff it is outer and its radial boundary zero set has logarithmic capacity zero (Brown and Shields 1984). For a nonzero Hardy space function, outerness means equality in the logarithmic mean formula

The radial zero set consists of those on the unit circle for which the full complex radial limit exists and equals zero.

Chen et al. (2026) report a counterexample: an outer with radial zero set of logarithmic capacity zero whose polynomial multiples do not approximate 1. Their manuscript explicitly leaves the complete proof and its principal ingredients subject to independent review, so this is a claimed refutation. In their normalized construction, and

The constant polynomial attains this minimum. The claimed construction combines approximation by canonical generators associated with finite atomic singular inner functions with recovery by outer generators having boundary zero sets of shrinking capacity (Chen et al. 2026).