  We consider local well-posedness for the Maxwell-Chern-Simons-Higgs system in
Lorenz gauge for data with minimal regularity assumptions in Fourier-Lebesgue
spaces $\widehat{H}^{s,r}$ , where $\|u\|_{\widehat{H}^{s,r}} := \| \langle \xi
\rangle^s \widehat{u}(\xi)\|_{L^{r'}}$ , and $r$ and $r'$ are dual exponents.
We show that the gap between this regularity and the regularity with respect to
scaling shrinks in the case $r>1$ , $r \to 1$ compared to the classical case
$r=2$ .
