  We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ of any dimension
(including $1$) which do not contain any arithmetic progressions of length $3$.
Moreover, the sets can be taken to be Ahlfors regular if the dimension is less
than $1$, and the measure witnessing the Fourier decay can be taken to be
Frostman in the case of dimension $1$. This is in sharp contrast to the
situation in the discrete setting (where Fourier uniformity is well known to
imply existence of progressions), and helps clarify a result of Laba and
Pramanik on pseudo-random subsets of the real line which do contain
progressions.
