The Ionescu--Wainger multiplier theorem and the adeles

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 ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$ norms of functions with Fourier transform supported on major arcs are comparable to the $L^p({\text{A}}_{\mathbb{Z}}^d)$ norm of their adelic counterparts.