Fourier-invariant functions with dense zero sets

Andriy Bondarenko, Kristian Seip

For every $0\le \beta \le 1/2$, we construct a nonzero real-valued continuous function $f_{\beta }$ in $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ such that ${\hat{f}}_{\beta }=f_{\beta }$ and $f_β(\sqrt{n}/[\lo…