Bohr sets for multiplier algebras of complete Pick spaces

We reexamine F. Wiener’s classical proof that the Bohr radius of $H^{\mathrm{\infty }}$ is $1\mathrm{/}3$. We replace $H^{\mathrm{\infty }}$ by the multiplier algebras of complete Pick spaces and describe the Bohr set for a wide class of such algebras. In particular, we show that the Bohr set of the multiplier algebra of the Drury–Arveson space $H_d^2$ is the closed ${\mathrm{\ell }}^2$-ball of radius $1\mathrm{/}3$. Other cases covered by our description of Bohr sets include Dirichlet-type spaces of analytic functions in the unit disc and a class of Hilbert spaces of ordinary Dirichlet series, studied earlier by McCarthy and Shalit.