We study the regularity of convolution powers for measures supported on Salem sets, and
prove related results on Fourier restriction and Fourier multipliers. In particular we show that for
$\alpha $
of the form
$d\,/\,n,\,n\,=\,2,3,...$
there exist
$\alpha $
-Salem measures for which the
${{L}^{2}}$
Fourier restriction theorem holds in the range
$p\,\le \,\frac{2d}{2d\,-\,\alpha }$
. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular
$\alpha $
-Salem measures, with sharp regularity results for
$n$
-fold
convolutions for all
$n\,\in \,\mathbb{N}$
.