Variants of a theorem of Helson on general Dirichlet series (Q2182599)

The authors started developing a theory of Hardy spaces of general Dirichlet series, analogous to that of Bayart for ordinary Dirichlet series, in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No. 6, 3220--3258 (2019; Zbl 1429.43004)]. Roughly speaking, given a frequency \(\lambda = (\lambda_{n})_{n}\) (i.e., an increasing sequence of non-negative numbers tending to \(\infty\)), the Hardy space \(\mathcal{H}_{p}(\lambda)\) (for \(1 \leq p \leq \infty\)) of general Dirichlet series \(\sum a_{n} e^{- \lambda_{n} s}\) is defined by means of a \(\lambda\)-Dirichlet group \((G, \beta)\) (which in particular is a compact, abelian group). The programme of developing a general theory of Hardy spaces of general Dirichlet series is continued here, giving a generalisation of a result of Helson about the convergence of vertical limits of general Dirichlet series. Given a general Dirichlet series \(D = \sum a_{n} e^{- \lambda_{n} s}\), a \textit{vertical limit} is a series of the form \[ D^{\omega} = \sum a_{n} h_{\lambda_{n}}(\omega) e^{- \lambda_{n} s} \,, \] for \(\omega \in G\). \\ A frequency \(\lambda\) is said to satisfy \textit{Landau's condition} (LC) if for every \(\delta > 0\) there is \(C > 0\) so that \(\lambda_{n+1}- \lambda_{n} \geq C e^{-e^{\delta n}} \), for every \(n \in \mathbb{N}\). The main result of the paper shows the following: \begin{itemize} \item If \(1 < p < \infty\) and \(D \in \mathcal{H}_{p}(\lambda)\), then for almost every \(\omega \in G\), the vertical limit \(D^{\omega}\) converges almost everywhere on \([\Re s=0]\) and, in particular, almost every \(D^{\omega}\) converges on \([\Re s>0]\). \item If \(\lambda\) satisfies (LC) and \(D \in \mathcal{H}_{1}(\lambda)\), then for almost every \(\omega \in G\), the vertical limit \(D^{\omega}\) converges on \([\Re s>0]\). \end{itemize} The proof relies heavily on the relationship with general Dirichlet series and the functions in the Hardy space \(H_{p}^{\lambda}(G)\). The reflexive case (for \(1 <p<\infty\)) is based on a Carleson-Hunt maximal inequality for \(H_{p}^{\lambda}(G)\). The case \(p=1\) follows from another maximal inequality for functions in \(H_{1}^{\lambda}(G)\) with \(\lambda\) satisfying (LC). As a consequence, it is shown that the space \(\mathcal{D}_{\infty}(\lambda)\) (defined as that of \(\lambda\)-Dirichlet series which converge on \([\Re s>0]\) and define a bounded holomorphic function there) is complete if and only if it is isometrically isomorphic to \(\mathcal{H}_{\infty}(\lambda)\), and this is equivalent to \(\lambda\) satisfying Bohr's theorem. Also, bounds for the norm of the partial sum operator for general Dirichlet series are obtained.