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A result of Helson on general Dirichlet series $\sum a_{n} e^{-\lambda_{n}s}$states that, whenever $(a_{n})$ is $2$-summable and $\lambda=(\lambda_{n})$satisfies a certain condition introduced by Bohr, then for almost allhomomorphism $\omega \colon (\mathbb{R},+) \to \mathbb{T}$ the Dirichlet series$\sum a_{n} \omega(\lambda_{n})e^{-\lambda_{n}s}$ converges on the open righthalf plane $[Re>0]$. For ordinary Dirichlet series $\sum a_n n^{-s}$ Hedenmalmand Saksman related this result with the famous Carleson-Hunt theorem onpointwise convergence of Fourier series, and Bayart extended it within histheory of Hardy spaces $\mathcal{H}_p$ of such series. The aim here is to provevariants of Helson's theorem within our recent theory of Hardy spaces$\mathcal{H}_{p}(\lambda),\,1\le p \le \infty,$ of general Dirichlet series. Tobe more precise, in the reflexive case $1 < p < \infty$ we extend Helson'sresult to Dirichlet series in $\mathcal{H}_{p}(\lambda)$ without any furthercondition on the frequency $\lambda$, and in the non-reflexive case $p=1$ tothe wider class of frequencies satisfying the so-called Landau condition (moregeneral than Bohr's condition). In both cases we add relevant maximalinequalities. Finally, we give several applications to the structure theory ofHardy spaces of general Dirichlet series.

Variants of a theorem of Helson on general Dirichlet series