  The Ionescu--Wainger multiplier theorem establishes good $L^p$ bounds for
Fourier multiplier operators localized to major arcs; it has become an
indispensible tool in discrete harmonic analysis. We give a simplified proof of
this theorem with more explicit constants (removing logarithmic losses that
were present in previous versions of the theorem), and give a more general
variant involving adelic Fourier multipliers. We also establish a closely
related adelic sampling theorem that shows that $\ell^p(\Z^d)$ norms of
functions with Fourier transform supported on major arcs are comparable to the
$L^p(\A_\Z^d)$ norm of their adelic counterparts.
